2.17.41 Problems 4001 to 4100

Problem 4001

ODE	\[ \boxed {{y^{\prime }}^{2}-y^{2} x^{2}=0} \]
program solution	\[ y = c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ y \left (x \right ) &= {\mathrm e}^{-\frac {x^{2}}{2}} c_{1} \\ \end{align*}

Problem 4002

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (y-1\right ) y^{2}=0} \]
program solution	\[ 2 \arctan \left (\sqrt {y-1}\right ) = x +c_{1} \] Verified OK. \[ -2 \arctan \left (\sqrt {y-1}\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 1 \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sec \left (\frac {c_{1}}{2}-\frac {x}{2}\right )^{2} \\ \end{align*}

Problem 4003

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (y-a \right ) \left (y-b \right ) \left (y-c \right )=0} \]
program solution	\[ \frac {2 \left (-a +b \right ) \sqrt {\frac {y-a}{-a +b}}\, \sqrt {\frac {y-c}{a -c}}\, \sqrt {\frac {y-b}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {y-a}{-a +b}}, \sqrt {\frac {a -b}{a -c}}\right )}{\sqrt {y^{3}-y^{2} a -b y^{2}-y^{2} c +y a b +y a c +y b c -a b c}} = x +c_{1} \] Verified OK. \[ -\frac {2 \left (-a +b \right ) \sqrt {\frac {y-a}{-a +b}}\, \sqrt {\frac {y-c}{a -c}}\, \sqrt {\frac {y-b}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {y-a}{-a +b}}, \sqrt {\frac {a -b}{a -c}}\right )}{\sqrt {y^{3}-y^{2} a -b y^{2}-y^{2} c +y a b +y a c +y b c -a b c}} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ y \left (x \right ) &= c \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-a +\textit {\_a} \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-a +\textit {\_a} \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*}

Problem 4004

ODE	\[ \boxed {{y^{\prime }}^{2}-a^{2} y^{n}=0} \]
program solution	\[ -\frac {2 y}{\left (n -2\right ) \sqrt {y^{n}}\, a} = x +c_{1} \] Verified OK. \[ \frac {2 y}{\left (n -2\right ) \sqrt {y^{n}}\, a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 4^{\frac {1}{n -2}} \left (-\frac {1}{a \left (-c_{1} +x \right ) \left (n -2\right )}\right )^{\frac {2}{n -2}} \\ y \left (x \right ) &= 4^{\frac {1}{n -2}} \left (\frac {1}{a \left (-c_{1} +x \right ) \left (n -2\right )}\right )^{\frac {2}{n -2}} \\ \end{align*}

Problem 4005

ODE	\[ \boxed {{y^{\prime }}^{2}-a^{2} \left (1-\ln \left (y\right )^{2}\right ) y^{2}=0} \]
program solution	\[ y = {\mathrm e}^{\sin \left (a c_{1} +a x \right )} \] Verified OK. \[ y = {\mathrm e}^{-\sin \left (a \left (x +c_{2} \right )\right )} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= {\mathrm e}^{\operatorname {RootOf}\left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )} \\ y \left (x \right ) &= {\mathrm e}^{-\sin \left (\left (c_{1} -x \right ) a \right )} \\ y \left (x \right ) &= {\mathrm e}^{\sin \left (\left (c_{1} -x \right ) a \right )} \\ \end{align*}

Problem 4006

ODE	\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right )=0} \]
program solution	\[ y = \frac {\left (4 \,{\mathrm e}^{\int 2 \sqrt {-f \left (x \right )}d x +2 c_{1}}+4 \,{\mathrm e}^{\int \sqrt {-f \left (x \right )}d x +c_{1}} a +4 \,{\mathrm e}^{\int \sqrt {-f \left (x \right )}d x +c_{1}} b +a^{2}-2 a b +b^{2}\right ) {\mathrm e}^{\int -\sqrt {-f \left (x \right )}d x -c_{1}}}{8} \] Verified OK. \[ y = \frac {\left (4 \,{\mathrm e}^{\int -2 \sqrt {-f \left (x \right )}d x -2 c_{1}}+4 \,{\mathrm e}^{-\left (\int \sqrt {-f \left (x \right )}d x \right )-c_{1}} a +4 \,{\mathrm e}^{-\left (\int \sqrt {-f \left (x \right )}d x \right )-c_{1}} b +a^{2}-2 a b +b^{2}\right ) {\mathrm e}^{\int \sqrt {-f \left (x \right )}d x +c_{1}}}{8} \] Verified OK.
Maple solution	\begin{align*} \frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \left (-\ln \left (2\right )+\ln \left (-a -b +2 y \left (x \right )+2 \sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\right )\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}-\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+c_{1} &= 0 \\ \frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \left (-\ln \left (2\right )+\ln \left (-a -b +2 y \left (x \right )+2 \sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\right )\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+c_{1} &= 0 \\ \end{align*}

Problem 4007

ODE	\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\left (a -y \right )^{2} \left (b -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (a-y)^2*(b-y), 0 < f(x)} \[ \int _{}^{y}-\frac {1}{\sqrt {\left (a -y \right )^{2} \left (b -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (a-y)^2*(b-y), 0 < f(x)}
Maple solution	\begin{align*} \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \end{align*}

Problem 4008

ODE	\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\left (c -y \right ) \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (c-y)*(b-y)*(a-y), 0 < f(x)} \[ \int _{}^{y}-\frac {1}{\sqrt {\left (c -y \right ) \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (c-y)*(b-y)*(a-y), 0 < f(x)}
Maple solution	\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-a +\textit {\_a} \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )}d \textit {\_a}}{\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-a +\textit {\_a} \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} +\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )}d \textit {\_a}}{\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}}+c_{1} &= 0 \\ \end{align*}

Problem 4009

ODE	\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\left (a -y \right )^{2} \left (c -y \right ) \left (b -y \right )}}d \textit {\_a} = \int \sqrt {-f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (a-y)^2*(c-y)*(b-y), 0 < -f(x)} \[ \int _{}^{y}-\frac {1}{\sqrt {\left (a -y \right )^{2} \left (c -y \right ) \left (b -y \right )}}d \textit {\_a} = \int \sqrt {-f \left (x \right )}d x +c_{1} \] Verified OK. {0 < (a-y)^2*(c-y)*(b-y), 0 < -f(x)}
Maple solution	\begin{align*} -\frac {\ln \left (\frac {-2 \sqrt {\left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}\, \sqrt {\left (a -b \right ) \left (a -c \right )}+\left (-2 a +b +c \right ) y \left (x \right )+\left (b +c \right ) a -2 b c}{-y \left (x \right )+a}\right ) \sqrt {a^{2}-a b -a c +b c}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (a -c \right ) \left (a -b \right ) \sqrt {b c -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}}+\frac {\int _{}^{x}\sqrt {\left (b -y \left (x \right )\right ) \left (y \left (x \right )-c \right ) f \left (\textit {\_a} \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ -\frac {\ln \left (\frac {-2 \sqrt {\left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}\, \sqrt {\left (a -b \right ) \left (a -c \right )}+\left (-2 a +b +c \right ) y \left (x \right )+\left (b +c \right ) a -2 b c}{-y \left (x \right )+a}\right ) \sqrt {a^{2}-a b -a c +b c}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (a -c \right ) \left (a -b \right ) \sqrt {b c -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}}-\frac {\int _{}^{x}\sqrt {\left (b -y \left (x \right )\right ) \left (y \left (x \right )-c \right ) f \left (\textit {\_a} \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \end{align*}

Problem 4010

ODE	\[ \boxed {{y^{\prime }}^{2}-f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\left (c -y \right )^{2} \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )^{2}}d x +c_{1} \] Verified OK. {0 < f(x)^2, 0 < (c-y)^2*(b-y)*(a-y)} \[ \int _{}^{y}-\frac {1}{\sqrt {\left (c -y \right )^{2} \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )^{2}}d x +c_{1} \] Verified OK. {0 < f(x)^2, 0 < (c-y)^2*(b-y)*(a-y)}
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c \,{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+\left (\left (4 b -2 c \right ) a -2 b c \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+c \left (a -b \right )^{2}}{{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+\left (2 a +2 b -4 c \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+a^{2}-2 a b +b^{2}} \\ y \left (x \right ) &= \frac {\left (\left (4 b -2 c \right ) a -2 b c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+\left ({\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+\left (a -b \right )^{2}\right ) c}{\left (2 a +2 b -4 c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+a^{2}-2 a b +b^{2}+{\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}} \\ \end{align*}

Problem 4011

ODE	\[ \boxed {{y^{\prime }}^{2}-f \left (x \right )^{2} \left (y-u \left (x \right )\right )^{2} \left (y-a \right ) \left (y-b \right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4012

ODE	\[ \boxed {{y^{\prime }}^{2}+2 y^{\prime }=-x} \]
program solution	\[ y = -x -\frac {2 \left (1-x \right )^{\frac {3}{2}}}{3}+c_{1} \] Verified OK. \[ y = -x +\frac {2 \left (1-x \right )^{\frac {3}{2}}}{3}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-2 x +2\right ) \sqrt {1-x}}{3}-x +c_{1} \\ y \left (x \right ) &= \frac {\left (2 x -2\right ) \sqrt {1-x}}{3}-x +c_{1} \\ \end{align*}

Problem 4013

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime }+a \left (-y+x \right )=0} \]
program solution	\[ y = \frac {a x -1}{a} \] Verified OK. \[ y = x +\frac {\left (\frac {a x}{2}+c_{1} \right ) \left (\frac {a x}{2}+c_{1} -2\right )}{a} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {a x -1}{a} \\ y \left (x \right ) &= \frac {\left (-c_{1} +x \right )^{2} a}{4}+c_{1} \\ \end{align*}

Problem 4014

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime }-y^{2}=0} \]
program solution	\[ -\frac {\left (1+y^{2}\right )^{\frac {3}{2}}}{y}+y \sqrt {1+y^{2}}+\operatorname {arcsinh}\left (y\right )+\frac {1}{y} = x +c_{1} \] Verified OK. \[ \frac {1}{y}+\frac {\left (1+y^{2}\right )^{\frac {3}{2}}}{y}-y \sqrt {1+y^{2}}-\operatorname {arcsinh}\left (y\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} \frac {-\sqrt {y \left (x \right )^{2}+1}+\operatorname {arcsinh}\left (y \left (x \right )\right ) y \left (x \right )-1+\left (-c_{1} +x \right ) y \left (x \right )}{y \left (x \right )} &= 0 \\ \frac {\sqrt {y \left (x \right )^{2}+1}-\operatorname {arcsinh}\left (y \left (x \right )\right ) y \left (x \right )-1+\left (-c_{1} +x \right ) y \left (x \right )}{y \left (x \right )} &= 0 \\ \end{align*}

Problem 4015

ODE	\[ \boxed {{y^{\prime }}^{2}-5 y^{\prime }=-6} \]
program solution	\[ y = 3 x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 3 x +c_{1} \\ y \left (x \right ) &= 2 x +c_{1} \\ \end{align*}

Problem 4016

ODE	\[ \boxed {{y^{\prime }}^{2}-7 y^{\prime }=-12} \]
program solution	\[ y = 4 x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 4 x +c_{1} \\ y \left (x \right ) &= 3 x +c_{1} \\ \end{align*}

Problem 4017

ODE	\[ \boxed {{y^{\prime }}^{2}+a y^{\prime }=-b} \]
program solution	\[ y = \frac {x \left (-a +\sqrt {a^{2}-4 b}\right )}{2}+c_{1} \] Verified OK. \[ y = x \left (-\frac {a}{2}-\frac {\sqrt {a^{2}-4 b}}{2}\right )+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {a x}{2}-\frac {x \sqrt {a^{2}-4 b}}{2}+c_{1} \\ y \left (x \right ) &= -\frac {a x}{2}+\frac {x \sqrt {a^{2}-4 b}}{2}+c_{1} \\ \end{align*}

Problem 4018

ODE	\[ \boxed {{y^{\prime }}^{2}+a y^{\prime }=-b x} \]
program solution	\[ y = -\frac {\left (a^{2}-4 b x \right )^{\frac {3}{2}}}{12 b}-\frac {a x}{2}+c_{1} \] Verified OK. \[ y = -\frac {a x}{2}+\frac {\left (a^{2}-4 b x \right )^{\frac {3}{2}}}{12 b}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (a^{2}-4 b x \right )^{\frac {3}{2}}-6 b \left (a x -2 c_{1} \right )}{12 b} \\ y \left (x \right ) &= \frac {\left (-a^{2}+4 b x \right ) \sqrt {a^{2}-4 b x}-6 b \left (a x -2 c_{1} \right )}{12 b} \\ \end{align*}

Problem 4019

ODE	\[ \boxed {{y^{\prime }}^{2}+a y^{\prime }+b y=0} \]
program solution	\[ -\frac {\sqrt {a^{2}-4 b y}+a \ln \left (a -\sqrt {a^{2}-4 b y}\right )}{b} = x +c_{1} \] Verified OK. \[ -\frac {-\sqrt {a^{2}-4 b y}+a \ln \left (a +\sqrt {a^{2}-4 b y}\right )}{b} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= -\frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (c_{1} -x \right ) b}{a}} \left (a \sqrt {-\frac {1}{b}}+{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (c_{1} -x \right ) b}{a}}\right ) \\ \end{align*}

Problem 4020

ODE	\[ \boxed {{y^{\prime }}^{2}+x y^{\prime }=-1} \]
program solution	\[ y = \frac {x \sqrt {x^{2}-4}}{4}-\ln \left (x +\sqrt {x^{2}-4}\right )-\frac {x^{2}}{4}+c_{1} \] Verified OK. \[ y = -\frac {x^{2}}{4}-\frac {x \sqrt {x^{2}-4}}{4}+\ln \left (x +\sqrt {x^{2}-4}\right )+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{4}-\frac {x \sqrt {x^{2}-4}}{4}+\ln \left (x +\sqrt {x^{2}-4}\right )+c_{1} \\ y \left (x \right ) &= \frac {x \sqrt {x^{2}-4}}{4}-\ln \left (x +\sqrt {x^{2}-4}\right )-\frac {x^{2}}{4}+c_{1} \\ \end{align*}

Problem 4021

ODE	\[ \boxed {{y^{\prime }}^{2}+x y^{\prime }-y=0} \]
program solution	\[ y = c_{1}^{2}+c_{1} x \] Verified OK. \[ y = -\frac {x^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (c_{1} +x \right ) \\ \end{align*}

Problem 4022

ODE	\[ \boxed {{y^{\prime }}^{2}-x y^{\prime }+y=0} \]
program solution	\[ y = -c_{1}^{2}+c_{1} x \] Verified OK. \[ y = \frac {x^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (-c_{1} +x \right ) \\ \end{align*}

Problem 4023

ODE	\[ \boxed {{y^{\prime }}^{2}-x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {x}{3}+\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}} \] Verified OK. \[ x = \frac {x}{3}-\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y \left (x \right )}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{3} &= 0 \\ \frac {c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y \left (x \right )}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}+4 y \left (x \right )}}{3} &= 0 \\ \end{align*}

Problem 4024

ODE	\[ \boxed {{y^{\prime }}^{2}+x y^{\prime }-y=-x} \]
program solution	\[ x = x -\sqrt {x^{2}+4 y-4 x}+2+c_{1} {\mathrm e}^{\frac {x}{2}-\frac {\sqrt {x^{2}+4 y-4 x}}{2}} \] Verified OK. \[ x = x +\sqrt {x^{2}+4 y-4 x}+2+c_{1} {\mathrm e}^{\frac {x}{2}+\frac {\sqrt {x^{2}+4 y-4 x}}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {x^{2}}{4}+\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-1+\frac {x}{2}}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-1+\frac {x}{2}}}{2}\right )+1 \]

Problem 4025

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (1-x \right ) y^{\prime }+y=0} \]
program solution	\[ y = -c_{1}^{2}+c_{1} x -c_{1} \] Verified OK. \[ y = \frac {\left (-1+x \right )^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x -1\right )^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (-c_{1} +x -1\right ) \\ \end{align*}

Problem 4026

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (x +1\right ) y^{\prime }+y=0} \]
program solution	\[ y = -c_{1}^{2}+c_{1} x +c_{1} \] Verified OK. \[ y = \frac {\left (x +1\right )^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x +1\right )^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (x +1-c_{1} \right ) \\ \end{align*}

Problem 4027

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (2-x \right ) y^{\prime }-y=-1} \]
program solution	\[ y = c_{1}^{2}+c_{1} x -2 c_{1} +1 \] Verified OK. \[ y = x -\frac {1}{4} x^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x -\frac {1}{4} x^{2} \\ y \left (x \right ) &= 1+c_{1}^{2}+c_{1} \left (-2+x \right ) \\ \end{align*}

Problem 4028

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (a +x \right ) y^{\prime }-y=0} \]
program solution	\[ y = a c_{1} +c_{1}^{2}+c_{1} x \] Verified OK. \[ y = -\frac {\left (a +x \right )^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (x +a \right )^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (c_{1} +a +x \right ) \\ \end{align*}

Problem 4029

ODE	\[ \boxed {{y^{\prime }}^{2}-2 x y^{\prime }=-1} \]
program solution	\[ y = \frac {x^{2}}{2}+\frac {x \sqrt {x^{2}-1}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{1} \] Verified OK. \[ y = \frac {x^{2}}{2}-\frac {x \sqrt {x^{2}-1}}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}-\frac {\sqrt {x^{2}-1}\, x}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {x^{2}}{2}+\frac {\sqrt {x^{2}-1}\, x}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{1} \\ \end{align*}

Problem 4030

ODE	\[ \boxed {{y^{\prime }}^{2}+2 x y^{\prime }=3 x^{2}} \]
program solution	\[ y = \frac {x^{2}}{2}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= -\frac {3 x^{2}}{2}+c_{1} \\ \end{align*}

Problem 4031

ODE	\[ \boxed {{y^{\prime }}^{2}+2 x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x^{2}-x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}+3 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+x^{2}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+x^{2}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 4032

ODE	\[ \boxed {{y^{\prime }}^{2}+2 x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x^{2}-x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}+3 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+x^{2}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+x^{2}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 4033

ODE	\[ \boxed {{y^{\prime }}^{2}-2 x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x -\frac {1}{2} c_{1}^{2} \] Verified OK. \[ y = \frac {x^{2}}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2} \\ y \left (x \right ) &= -\frac {c_{1} \left (-2 x +c_{1} \right )}{2} \\ \end{align*}

Problem 4034

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (1+2 x \right ) y^{\prime }=x \left (1-x \right )} \]
program solution	\[ y = \frac {x}{2}+\frac {\left (8 x +1\right )^{\frac {3}{2}}}{24}+\frac {x^{2}}{2}+c_{1} \] Verified OK. \[ y = \frac {x^{2}}{2}+\frac {x}{2}-\frac {\left (8 x +1\right )^{\frac {3}{2}}}{24}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-8 x -1\right ) \sqrt {8 x +1}}{24}+\frac {x^{2}}{2}+\frac {x}{2}+c_{1} \\ y \left (x \right ) &= \frac {x}{2}+\frac {\left (8 x +1\right )^{\frac {3}{2}}}{24}+\frac {x^{2}}{2}+c_{1} \\ \end{align*}

Problem 4035

ODE	\[ \boxed {{y^{\prime }}^{2}+2 \left (1-x \right ) y^{\prime }+2 y=2 x} \]
program solution	\[ x = -1+x +\sqrt {x^{2}-2 y+1}+c_{1} {\mathrm e}^{1-x -\sqrt {x^{2}-2 y+1}} \] Verified OK. \[ x = -1+x -\sqrt {x^{2}-2 y+1}+c_{1} {\mathrm e}^{1-x +\sqrt {x^{2}-2 y+1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{2}-\frac {\operatorname {LambertW}\left (-{\mathrm e}^{-x} c_{1} \right )^{2}}{2}-\operatorname {LambertW}\left (-{\mathrm e}^{-x} c_{1} \right ) \]

Problem 4036

ODE	\[ \boxed {{y^{\prime }}^{2}+3 x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 4037

ODE	\[ \boxed {{y^{\prime }}^{2}-4 \left (x +1\right ) y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} x -\frac {1}{4} c_{1}^{2}+c_{1} \] Verified OK. \[ y = \left (x +1\right )^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \left (x +1\right )^{2} \\ y \left (x \right ) &= -\frac {c_{1} \left (-4 x +c_{1} -4\right )}{4} \\ \end{align*}

Problem 4038

ODE	\[ \boxed {{y^{\prime }}^{2}+a x y^{\prime }=b c \,x^{2}} \]
program solution	\[ y = \frac {x^{2} \left (-a +\sqrt {a^{2}+4 b c}\right )}{4}+c_{1} \] Verified OK. \[ y = -\frac {x^{2} \left (a +\sqrt {a^{2}+4 b c}\right )}{4}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2} \sqrt {a^{2}+4 b c}}{4}-\frac {a \,x^{2}}{4}+c_{1} \\ y \left (x \right ) &= -\frac {x^{2} \sqrt {a^{2}+4 b c}}{4}-\frac {a \,x^{2}}{4}+c_{1} \\ \end{align*}

Problem 4039

ODE	\[ \boxed {{y^{\prime }}^{2}-a x y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} x -\frac {c_{1}^{2}}{a} \] Verified OK. \[ y = \frac {a \,x^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {a \,x^{2}}{4} \\ y \left (x \right ) &= \frac {c_{1} \left (a x -c_{1} \right )}{a} \\ \end{align*}

Problem 4040

ODE	\[ \boxed {{y^{\prime }}^{2}+a x y^{\prime }+y c=-b \,x^{2}} \]
program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x^{2}}}\frac {2}{\sqrt {-4 \textit {\_a} c +a^{2}-4 b}-4 \textit {\_a} -a}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 4041

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (b x +a \right ) y^{\prime }-b y=-c} \]
program solution	\[ y = c_{1} x +\frac {a c_{1} +c_{1}^{2}+c}{b} \] Verified OK. \[ y = \frac {-x^{2} b^{2}-2 a b x -a^{2}+4 c}{4 b} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-b^{2} x^{2}-2 b x a -a^{2}+4 c}{4 b} \\ y \left (x \right ) &= \frac {c_{1}^{2}+\left (b x +a \right ) c_{1} +c}{b} \\ \end{align*}

Problem 4042

ODE	\[ \boxed {{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime }=0} \]
program solution	\[ y = \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2}{3} x^{3}-x^{2}+c_{1} \\ y \left (x \right ) &= c_{1} \\ \end{align*}

Problem 4043

ODE	\[ \boxed {{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y=0} \]
program solution	\[ \frac {\ln \left (y\right )}{4}-\frac {\ln \left (a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}+8 y}\right )}{4}+\frac {\ln \left (-a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}+8 y}\right )}{4} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{4}+\frac {\ln \left (a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}+8 y}\right )}{4}-\frac {\ln \left (-a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}+8 y}\right )}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {a \,x^{4}}{8} \\ y \left (x \right ) &= \frac {c_{1} \left (a \,x^{2}+2 c_{1} \right )}{a} \\ \end{align*}

Problem 4044

ODE	\[ \boxed {{y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y=0} \]
program solution	\[ \frac {\ln \left (y\right )}{4}+\frac {\ln \left (a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}-4 y}\right )}{4}-\frac {\ln \left (-a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}-4 y}\right )}{4} = \ln \left (x \right )+c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{4}-\frac {\ln \left (a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}-4 y}\right )}{4}+\frac {\ln \left (-a \,x^{2}+\sqrt {a}\, \sqrt {a \,x^{4}-4 y}\right )}{4} = -\ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {a \,x^{4}}{4} \\ y \left (x \right ) &= \frac {c_{1} \left (a \,x^{2}-c_{1} \right )}{a} \\ \end{align*}

Problem 4045

ODE	\[ \boxed {{y^{\prime }}^{2}+4 y^{\prime } x^{5}-12 y x^{4}=0} \]
program solution	\[ \frac {\ln \left (y\right )}{6}+\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{6}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}+\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{6}}{3} \\ y \left (x \right ) &= c_{1} x^{3}+\frac {3}{4} c_{1}^{2} \\ \end{align*}

Problem 4046

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } \cosh \left (x \right )=-1} \]
program solution	\[ y = -{\mathrm e}^{-x}+c_{1} \] Verified OK. \[ y = {\mathrm e}^{x}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -{\mathrm e}^{-x}+c_{1} \\ y \left (x \right ) &= {\mathrm e}^{x}+c_{1} \\ \end{align*}

Problem 4047

ODE	\[ \boxed {{y^{\prime }}^{2}+y^{\prime } y-x \left (y+x \right )=0} \]
program solution	\[ y = 1-x +c_{2} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= 1+{\mathrm e}^{-x} c_{1} -x \\ \end{align*}

Problem 4048

ODE	\[ \boxed {{y^{\prime }}^{2}-y^{\prime } y=-{\mathrm e}^{x}} \]
program solution	\[ y = \frac {\left (4 \,{\mathrm e}^{x} {\mathrm e}^{2 c_{1}}+1\right ) {\mathrm e}^{-c_{1}}}{2} \] Verified OK. \[ y = \frac {\left ({\mathrm e}^{2 c_{1}}+4 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-c_{1}}}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -2 \,{\mathrm e}^{\frac {x}{2}} \\ y \left (x \right ) &= 2 \,{\mathrm e}^{\frac {x}{2}} \\ y \left (x \right ) &= \frac {c_{1}^{2} {\mathrm e}^{x}+1}{c_{1}} \\ \end{align*}

Problem 4049

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+y x=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= {\mathrm e}^{-x} c_{1} \\ \end{align*}

Problem 4050

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } y=2 x} \]
program solution	\[ y = -\frac {1}{2} i-i x \] Verified OK. \[ y = \frac {1}{2} i+i x \] Verified OK. \[ x = \frac {\left (y+\sqrt {y^{2}+2 x}\right ) \left (\operatorname {arcsinh}\left (y+\sqrt {y^{2}+2 x}\right )+2 c_{1} \right )}{2 \sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}} \] Verified OK. \[ x = -\frac {\left (-y+\sqrt {y^{2}+2 x}\right ) \left (-\operatorname {arcsinh}\left (-y+\sqrt {y^{2}+2 x}\right )+2 c_{1} \right )}{2 \sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}} \] Verified OK.
Maple solution	\begin{align*} \frac {\frac {\left (y \left (x \right )-\sqrt {y \left (x \right )^{2}+2 x}\right ) \operatorname {arcsinh}\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+2 x}\right )}{2}+x \sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}-2 c_{1} y \left (x \right )+2 c_{1} \sqrt {y \left (x \right )^{2}+2 x}}{\sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}} &= 0 \\ \frac {\frac {\left (-y \left (x \right )-\sqrt {y \left (x \right )^{2}+2 x}\right ) \operatorname {arcsinh}\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+2 x}\right )}{2}+x \sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}+2 c_{1} y \left (x \right )+2 c_{1} \sqrt {y \left (x \right )^{2}+2 x}}{\sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}} &= 0 \\ \end{align*}

Problem 4051

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (1+2 y\right ) y^{\prime }+y \left (y-1\right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{-\frac {1}{2}-\textit {\_a} +\frac {\sqrt {8 \textit {\_a} +1}}{2}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {1}{-\frac {1}{2}-\textit {\_a} -\frac {\sqrt {8 \textit {\_a} +1}}{2}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} x +\frac {3 \ln \left (y \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}-c_{1} &= 0 \\ x +\frac {3 \ln \left (y \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}-c_{1} &= 0 \\ \end{align*}

Problem 4052

ODE	\[ \boxed {{y^{\prime }}^{2}-2 \left (-y+x \right ) y^{\prime }-4 y x=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-2 x}}{c_{2}^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x^{2}+c_{1} \\ y \left (x \right ) &= {\mathrm e}^{-2 x} c_{1} \\ \end{align*}

Problem 4053

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (1+4 y\right ) y^{\prime }+\left (1+4 y\right ) y=0} \]
program solution	\[ \ln \left (\sqrt {1+4 y}+1\right ) = x +c_{1} \] Verified OK. \[ \ln \left (\sqrt {1+4 y}-1\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -{\frac {1}{4}} \\ y \left (x \right ) &= -\frac {\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{c_{1} \sqrt {-{\mathrm e}^{-2 x} c_{1}}} \\ y \left (x \right ) &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, c_{1}} \\ y \left (x \right ) &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{c_{1} \sqrt {-{\mathrm e}^{-2 x} c_{1}}} \\ \end{align*}

Problem 4054

ODE	\[ \boxed {{y^{\prime }}^{2}-2 \left (1-3 y\right ) y^{\prime }-\left (4-9 y\right ) y=0} \]
program solution	\[ -\frac {\ln \left (3 \sqrt {-2 y+1}-1\right )}{6}-\frac {\ln \left (\sqrt {-2 y+1}+1\right )}{2} = x +c_{1} \] Verified OK. \[ -\frac {\ln \left (3 \sqrt {-2 y+1}+1\right )}{6}-\frac {\ln \left (\sqrt {-2 y+1}-1\right )}{2} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= {\frac {4}{9}} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (\textit {\_Z}^{8} {\mathrm e}^{24 x}+24 \textit {\_Z}^{7} {\mathrm e}^{24 x}+240 \textit {\_Z}^{6} {\mathrm e}^{24 x}+1280 \textit {\_Z}^{5} {\mathrm e}^{24 x}+\left (3840 \,{\mathrm e}^{24 x}-1458 \,{\mathrm e}^{12 x} c_{1} \right ) \textit {\_Z}^{4}+\left (6144 \,{\mathrm e}^{24 x}+75816 \,{\mathrm e}^{12 x} c_{1} \right ) \textit {\_Z}^{3}+\left (4096 \,{\mathrm e}^{24 x}-209952 \,{\mathrm e}^{12 x} c_{1} \right ) \textit {\_Z}^{2}-23328 \textit {\_Z} \,{\mathrm e}^{12 x} c_{1} -11664 \,{\mathrm e}^{12 x} c_{1} +531441 c_{1}^{2}\right )}{9}+\frac {4}{9} \\ \end{align*}

Problem 4055

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (a +6 y\right ) y^{\prime }+y \left (3 a +b +9 y\right )=0} \]
program solution	\[ -\frac {2 a \ln \left (a -\sqrt {a^{2}-4 b y}\right )}{6 a +2 b}-\frac {2 \left (3 a +2 b \right ) \ln \left (3 a +2 b +3 \sqrt {a^{2}-4 b y}\right )}{3 \left (6 a +2 b \right )} = x +c_{1} \] Verified OK. \[ -\frac {2 \left (3 a +2 b \right ) \ln \left (3 a +2 b -3 \sqrt {a^{2}-4 b y}\right )}{3 \left (6 a +2 b \right )}-\frac {2 a \ln \left (a +\sqrt {a^{2}-4 b y}\right )}{6 a +2 b} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\operatorname {RootOf}\left (3 a \ln \left (-\frac {b}{\left (3 \textit {\_Z} -2 b \right )^{2}}\right )+2 b \ln \left (-\frac {b}{\left (3 \textit {\_Z} -2 b \right )^{2}}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )-3 a \ln \left (-\frac {\left (\textit {\_Z} +2 a \right )^{2}}{b}\right )+18 c_{1} a +6 c_{1} b -18 a x -6 b x \right ) \left (\operatorname {RootOf}\left (3 a \ln \left (-\frac {b}{\left (3 \textit {\_Z} -2 b \right )^{2}}\right )+2 b \ln \left (-\frac {b}{\left (3 \textit {\_Z} -2 b \right )^{2}}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )-3 a \ln \left (-\frac {\left (\textit {\_Z} +2 a \right )^{2}}{b}\right )+18 c_{1} a +6 c_{1} b -18 a x -6 b x \right )+2 a \right )}{4 b} \\ y \left (x \right ) &= -\frac {{\mathrm e}^{\operatorname {RootOf}\left (-3 a \ln \left (-\frac {\left (3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-2 b \ln \left (-\frac {\left (3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-3 a \ln \left (-\frac {1}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_{1} a +6 c_{1} b -6 a \textit {\_Z} -18 a x -6 b x \right )} \left ({\mathrm e}^{\operatorname {RootOf}\left (-3 a \ln \left (-\frac {\left (3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-2 b \ln \left (-\frac {\left (3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-3 a \ln \left (-\frac {1}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_{1} a +6 c_{1} b -6 a \textit {\_Z} -18 a x -6 b x \right )}+2 a \right )}{4 b} \\ \end{align*}

Problem 4056

ODE	\[ \boxed {{y^{\prime }}^{2}+a y y^{\prime }=a x} \]
program solution	\[ y = -\frac {a x -1}{a} \] Verified OK. \[ y = \frac {a x -1}{a} \] Verified OK. \[ x = \left (-a y+\sqrt {a \left (y^{2} a +4 x \right )}\right ) \left (-\frac {-\ln \left (2\right )+\ln \left (-a y+\sqrt {2 a^{2} y^{2}-2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x -4}+\sqrt {a \left (y^{2} a +4 x \right )}\right )}{\sqrt {2 a^{2} y^{2}-2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x -4}\, a}+\frac {2 c_{1}}{\sqrt {-2 a y+2 \sqrt {a \left (y^{2} a +4 x \right )}-4}\, \sqrt {-2 a y+2 \sqrt {a \left (y^{2} a +4 x \right )}+4}}\right ) \] Verified OK. \[ x = \left (a y+\sqrt {a \left (y^{2} a +4 x \right )}\right ) \left (-\frac {\ln \left (2\right )-\ln \left (-a y-\sqrt {a \left (y^{2} a +4 x \right )}+\sqrt {2 a^{2} y^{2}+2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x -4}\right )}{\sqrt {2 a^{2} y^{2}+2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x -4}\, a}-\frac {2 c_{1}}{\sqrt {-2 a y-2 \sqrt {a \left (y^{2} a +4 x \right )}-4}\, \sqrt {-2 a y-2 \sqrt {a \left (y^{2} a +4 x \right )}+4}}\right ) \] Verified OK.
Maple solution	\begin{align*} \frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}\, \sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}}+x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (-\ln \left (2\right )+\ln \left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+\sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x -4}\right )\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x -4}} &= 0 \\ \frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}\, \sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}}+x -\frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (-\ln \left (2\right )+\ln \left (-a y \left (x \right )-\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+\sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x -4}\right )\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x -4}} &= 0 \\ \end{align*}

Problem 4057

ODE	\[ \boxed {{y^{\prime }}^{2}-a y y^{\prime }=a x} \]
program solution	\[ y = \frac {-i a x -i}{a} \] Verified OK. \[ y = \frac {i a x +i}{a} \] Verified OK. \[ x = \frac {\left (a y+\sqrt {a \left (y^{2} a +4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y}{2}+\frac {\sqrt {a \left (y^{2} a +4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y^{2}+2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x +4}\, a} \] Verified OK. \[ x = \frac {\left (a y-\sqrt {a \left (y^{2} a +4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y}{2}-\frac {\sqrt {a \left (y^{2} a +4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y^{2}-2 a y \sqrt {a \left (y^{2} a +4 x \right )}+4 a x +4}\, a} \] Verified OK.
Maple solution	\begin{align*} x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}-\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x +4}\, a} &= 0 \\ x -\frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}+\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\, a y \left (x \right )+4 a x +4}\, a} &= 0 \\ \end{align*}

Problem 4058

ODE	\[ \boxed {{y^{\prime }}^{2}+\left (a x +b y\right ) y^{\prime }+a b x y=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-b x}}{c_{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {a \,x^{2}}{2}+c_{1} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-b x} \\ \end{align*}

Problem 4059

ODE	\[ \boxed {{y^{\prime }}^{2}-y y^{\prime } x +y^{2} \ln \left (a y\right )=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-c_{1}^{2}+c_{1} x}}{a} \] Verified OK. \[ y = \frac {{\mathrm e}^{-c_{1}^{2}+c_{1} x}}{a} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{a} \\ y \left (x \right ) &= \frac {{\mathrm e}^{c_{1} \left (-c_{1} +x \right )}}{a} \\ y \left (x \right ) &= \frac {{\mathrm e}^{-c_{1} \left (c_{1} +x \right )}}{a} \\ \end{align*}

Problem 4060

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (1+2 y x \right ) y^{\prime }+2 y x=0} \]
program solution	\[ y = c_{2} {\mathrm e}^{x^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} +x \\ y \left (x \right ) &= {\mathrm e}^{x^{2}} c_{1} \\ \end{align*}

Problem 4061

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (4+y^{2}\right ) y^{\prime }+y^{2}=-4} \]
program solution	\[ \frac {y}{2}-\frac {1}{2}-\frac {\sqrt {y^{4}+4 y^{2}}}{2 y} = x +c_{1} \] Verified OK. \[ \frac {y}{2}-\frac {1}{2}+\frac {\sqrt {y^{4}+4 y^{2}}}{2 y} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -2 i \\ y \left (x \right ) &= 2 i \\ x +2 \left (\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}-4}d \textit {\_a} \right )-c_{1} &= 0 \\ x -2 \left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}+4}d \textit {\_a} \right )-c_{1} &= 0 \\ \end{align*}

Problem 4062

ODE	\[ \boxed {{y^{\prime }}^{2}-\left (-y+x \right ) y y^{\prime }-x y^{3}=0} \]
program solution	\[ y = c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1}{c_{1} +x} \\ y \left (x \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ \end{align*}

Problem 4063

ODE	\[ \boxed {{y^{\prime }}^{2}+y^{2} y^{\prime } x +y^{3}=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{2}-\frac {\ln \left (y x^{2}+x \sqrt {y}\, \sqrt {y x^{2}-4}-2\right )}{2}-c_{1} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {4}{x^{2}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {2 \sqrt {2}\, x -2 c_{1}}{c_{1} \left (c_{1}^{2}-2 x^{2}\right )} \\ y \left (x \right ) &= \frac {-2 \sqrt {2}\, x -2 c_{1}}{c_{1} \left (c_{1}^{2}-2 x^{2}\right )} \\ y \left (x \right ) &= -\frac {\left (\sqrt {2}\, c_{1} x -2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{2}-4} \\ y \left (x \right ) &= \frac {\left (\sqrt {2}\, c_{1} x +2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{2}-4} \\ \end{align*}

Problem 4064

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } y^{2} x^{3}-4 y^{3} x^{2}=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{4}+\frac {\ln \left (y x^{4}+x^{2} \sqrt {y}\, \sqrt {y x^{4}+4}+2\right )}{4}-c_{1} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {4}{x^{4}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (\sqrt {2}\, x^{2} c_{1} -2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{4}-4} \\ y \left (x \right ) &= -\frac {\left (\sqrt {2}\, x^{2} c_{1} +2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{4}-4} \\ y \left (x \right ) &= \frac {-2 \sqrt {2}\, x^{2}+2 c_{1}}{c_{1} \left (-2 x^{4}+c_{1}^{2}\right )} \\ y \left (x \right ) &= \frac {2 \sqrt {2}\, x^{2}+2 c_{1}}{c_{1} \left (-2 x^{4}+c_{1}^{2}\right )} \\ \end{align*}

Problem 4065

ODE	\[ \boxed {{y^{\prime }}^{2}-x y \left (x^{2}+y^{2}\right ) y^{\prime }+y^{4} x^{4}=0} \]
program solution	\[ y = -\frac {1}{\sqrt {-x^{2}-2 c_{2}}} \] Verified OK. \[ y = \frac {1}{\sqrt {-x^{2}-2 c_{2}}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {-x^{2}+c_{1}}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-x^{2}+c_{1}}} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{\frac {x^{4}}{4}} \\ \end{align*}

Problem 4066

ODE	\[ \boxed {{y^{\prime }}^{2}+2 y^{\prime } y^{3} x +y^{4}=0} \]
program solution	\[ \ln \left (y x +\sqrt {y^{2} x^{2}-1}\right )+\ln \left (y\right ) = c_{1} \] Verified OK. \[ \ln \left (y\right )-\ln \left (y x +\sqrt {y^{2} x^{2}-1}\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{x} \\ y \left (x \right ) &= \frac {1}{x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {1}{\sqrt {-c_{1} \left (-2 x +c_{1} \right )}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {c_{1} \left (-c_{1} +2 x \right )}} \\ \end{align*}

Problem 4067

ODE	\[ \boxed {{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2}=0} \]
program solution	\[ y = \frac {2 c_{1} \left (-\cot \left (x \right )+\csc \left (x \right )\right )^{\sqrt {\sec \left (x \right )^{2}}\, \cos \left (x \right )}}{\left (-\cot \left (x \right )+\csc \left (x \right )\right ) \left (\cos \left (x \right )+1\right )} \] Verified OK. \[ y = \frac {2 c_{2} \left (-\cot \left (x \right )+\csc \left (x \right )\right )^{-\sqrt {\sec \left (x \right )^{2}}\, \cos \left (x \right )}}{\left (-\cot \left (x \right )+\csc \left (x \right )\right ) \left (\cos \left (x \right )+1\right )} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1}}{\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )} \\ y \left (x \right ) &= \csc \left (x \right )^{2} \left (\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1} \\ \end{align*}

Problem 4068

ODE	\[ \boxed {{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}}=0} \]
program solution	\[ \ln \left (x \right ) = -i \arctan \left (\sqrt {4 \left (\frac {y}{x^{6}}\right )^{\frac {1}{3}}-1}\right )-\frac {\ln \left (\frac {y}{x^{6}}\right )}{6}+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{6}}{64} \\ y \left (x \right ) &= 0 \\ \ln \left (x \right )+\frac {\sqrt {-\frac {y \left (x \right ) \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}} \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{x^{6}}}\, \operatorname {arctanh}\left (\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}\right )}{\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}} \sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}}+\frac {\ln \left (\frac {64 y \left (x \right )}{x^{6}}-1\right )}{6}-\frac {\ln \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}}+4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\ln \left (\frac {y \left (x \right )}{x^{6}}\right )}{6}-c_{1} &= 0 \\ \end{align*}

Problem 4069

ODE	\[ \boxed {{y^{\prime }}^{2}-{\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right )=0} \]
program solution	\[ y = 2 x -\ln \left (2\right ) \] Verified OK. \[ y = 2 x -\frac {\ln \left (\frac {{\mathrm e}^{4 x} c_{1}^{4}}{\left (-1+{\mathrm e}^{2 x} c_{1}^{2}\right )^{2} \left (\frac {{\mathrm e}^{2 x} c_{1}^{2}}{-1+{\mathrm e}^{2 x} c_{1}^{2}}-1\right )}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} \frac {-\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{-2 x +y \left (x \right )}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\ \frac {\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{-2 x +y \left (x \right )}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\ \end{align*}

Problem 4070

ODE	\[ \boxed {2 {y^{\prime }}^{2}+x y^{\prime }-2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = -\frac {\left (x -\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x +\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK. \[ x = -\frac {\left (x +\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x -\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2} \left (1+2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )\right )}{16 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )^{2}} \]

Problem 4071

ODE	\[ \boxed {2 {y^{\prime }}^{2}-\left (1-x \right ) y^{\prime }-y=0} \]
program solution	\[ y = 2 c_{1}^{2}+c_{1} x -c_{1} \] Verified OK. \[ y = -\frac {\left (-1+x \right )^{2}}{8} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (x -1\right )^{2}}{8} \\ y \left (x \right ) &= c_{1} \left (2 c_{1} +x -1\right ) \\ \end{align*}

Problem 4072

ODE	\[ \boxed {2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 y x=0} \]
program solution	\[ \frac {\ln \left (y\right )}{3}-\frac {\ln \left (-x^{2}+\sqrt {x}\, \sqrt {x^{3}-6 y}\right )}{3}+\frac {\ln \left (x^{2}+\sqrt {x}\, \sqrt {x^{3}-6 y}\right )}{3} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{3}+\frac {\ln \left (-x^{2}+\sqrt {x}\, \sqrt {x^{3}-6 y}\right )}{3}-\frac {\ln \left (x^{2}+\sqrt {x}\, \sqrt {x^{3}-6 y}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{3}}{6} \\ y \left (x \right ) &= \frac {\sqrt {6}\, \sqrt {-c_{1} x}\, x}{3}+c_{1} \\ y \left (x \right ) &= -\frac {\sqrt {6}\, \sqrt {-c_{1} x}\, x}{3}+c_{1} \\ y \left (x \right ) &= -\frac {\sqrt {6}\, \sqrt {-c_{1} x}\, x}{3}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {6}\, \sqrt {-c_{1} x}\, x}{3}+c_{1} \\ \end{align*}

Problem 4073

ODE	\[ \boxed {2 {y^{\prime }}^{2}+2 \left (6 y-1\right ) y^{\prime }+3 y \left (6 y-1\right )=0} \]
program solution	\[ -\frac {2 \ln \left (\sqrt {-6 y+1}+1\right )}{3} = x +c_{1} \] Verified OK. \[ -\frac {2 \ln \left (\sqrt {-6 y+1}-1\right )}{3} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= {\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (\sqrt {6}\, {\mathrm e}^{\frac {3 x}{2}+\frac {3 c_{1}}{2}}+3 \,{\mathrm e}^{3 c_{1}}\right ) {\mathrm e}^{-3 x}}{3} \\ y \left (x \right ) &= \frac {\left (\sqrt {6}\, {\mathrm e}^{\frac {3 x}{2}+\frac {3 c_{1}}{2}}-3 \,{\mathrm e}^{3 c_{1}}\right ) {\mathrm e}^{-3 x}}{3} \\ \end{align*}

Problem 4074

ODE	\[ \boxed {3 {y^{\prime }}^{2}-2 x y^{\prime }+y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (8 x^{2}-6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x +\sqrt {x^{2}-3 y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (-8 x^{2}+6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x -\sqrt {x^{2}-3 y}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (x^{2}+x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}-3 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{12 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}-2 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+6 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{48 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, x^{2}-i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}-2 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}+6 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{48 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 4075

ODE	\[ \boxed {3 {y^{\prime }}^{2}+4 x y^{\prime }-y=-x^{2}} \]
program solution	\[ y = \frac {3}{16} c_{1}^{2}+\frac {1}{4} c_{1} x -\frac {1}{4} x^{2} \] Verified OK. \[ y = \frac {3}{16} c_{1}^{2}+\frac {1}{4} c_{1} x -\frac {1}{4} x^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{3} \\ y \left (x \right ) &= -\frac {x^{2}}{4}+\frac {\sqrt {3}\, c_{1} x}{6}+\frac {c_{1}^{2}}{4} \\ y \left (x \right ) &= -\frac {x^{2}}{4}-\frac {\sqrt {3}\, c_{1} x}{6}+\frac {c_{1}^{2}}{4} \\ y \left (x \right ) &= -\frac {x^{2}}{4}-\frac {\sqrt {3}\, c_{1} x}{6}+\frac {c_{1}^{2}}{4} \\ y \left (x \right ) &= -\frac {x^{2}}{4}+\frac {\sqrt {3}\, c_{1} x}{6}+\frac {c_{1}^{2}}{4} \\ \end{align*}

Problem 4076

ODE	\[ \boxed {4 {y^{\prime }}^{2}=9 x} \]
program solution	\[ y = x^{\frac {3}{2}}+c_{1} \] Verified OK. \[ y = -x^{\frac {3}{2}}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -x^{\frac {3}{2}}+c_{1} \\ y \left (x \right ) &= x^{\frac {3}{2}}+c_{1} \\ \end{align*}

Problem 4077

ODE	\[ \boxed {4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y}=0} \]
program solution	\[ \frac {i \pi }{2}-\ln \left (x \right )-\ln \left (\sqrt {4 \,{\mathrm e}^{2 y}+x^{2}}+x \right )+2 y = \ln \left (\frac {\tan \left (\operatorname {RootOf}\left (i \ln \left (\frac {2 x^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+2 i c_{1} +2 \textit {\_Z} \right )\right )}{2}\right ) \] Verified OK. \[ y = \frac {\ln \left (-\frac {{\mathrm e}^{2 c_{1}}}{64}+\frac {{\mathrm e}^{\frac {i \pi }{2}+c_{1}} x}{8}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\ln \left (2\right )-\frac {\ln \left (-\frac {1}{x^{2}}\right )}{2} \\ y \left (x \right ) &= c_{1} -\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}-x^{2}\right )}\right )}\right ) \\ y \left (x \right ) &= c_{1} +\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}-x^{2}\right )}\right )}\right ) \\ \end{align*}

Problem 4078

ODE	\[ \boxed {4 {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y}=0} \]
program solution	\[ y = x -\ln \left (2\right )-\frac {i \pi }{2} \] Verified OK. \[ y = x -\ln \left (2\right )-\frac {\ln \left (-\frac {{\mathrm e}^{2 x} c_{1}^{2}}{4 \left ({\mathrm e}^{x} c_{1} -1\right )^{2} \left (\frac {{\mathrm e}^{x} c_{1}}{{\mathrm e}^{x} c_{1} -1}-1\right )}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} -\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}+16 \,{\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}-8 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ y \left (x \right ) &= c_{1} +\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}+16 \,{\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}-8 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ \end{align*}

Problem 4079

ODE	\[ \boxed {5 {y^{\prime }}^{2}+3 x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 4080

ODE	\[ \boxed {5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 4081

ODE	\[ \boxed {9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5}=0} \]
program solution	\[ \ln \left (y\right )+\frac {\ln \left (x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3}-\frac {\ln \left (-x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )-\frac {\ln \left (x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3}+\frac {\ln \left (-x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}}}{x^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}+\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}-4}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} \right )+2 c_{1} \right )}{x^{\frac {2}{3}}} \\ \end{align*}

Problem 4082

ODE	\[ \boxed {x {y^{\prime }}^{2}=a} \]
program solution	\[ y = 2 \sqrt {a x}+c_{1} \] Verified OK. \[ y = -2 \sqrt {a x}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 2 \sqrt {a x}+c_{1} \\ y \left (x \right ) &= -2 \sqrt {a x}+c_{1} \\ \end{align*}

Problem 4083

ODE	\[ \boxed {x {y^{\prime }}^{2}=-x^{2}+a} \]
program solution	\[ y = \frac {2 \sqrt {-x^{3}+a x}}{3}+\frac {2 a^{\frac {3}{2}} \sqrt {\frac {x +\sqrt {a}}{\sqrt {a}}}\, \sqrt {-\frac {2 \left (x -\sqrt {a}\right )}{\sqrt {a}}}\, \sqrt {-\frac {x}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\sqrt {a}}{\sqrt {a}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-x^{3}+a x}}+c_{1} \] Verified OK. \[ y = -\frac {2 \sqrt {-x^{3}+a x}}{3}-\frac {2 a^{\frac {3}{2}} \sqrt {\frac {x +\sqrt {a}}{\sqrt {a}}}\, \sqrt {-\frac {2 \left (x -\sqrt {a}\right )}{\sqrt {a}}}\, \sqrt {-\frac {x}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\sqrt {a}}{\sqrt {a}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-x^{3}+a x}}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +c_{1} \\ y \left (x \right ) &= -\left (\int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x \right )+c_{1} \\ \end{align*}

Problem 4084

ODE	\[ \boxed {x {y^{\prime }}^{2}-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = x \] Verified OK. \[ y = x \left (1+\frac {c_{1}}{\sqrt {x}}\right )^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (x +\sqrt {c_{1} x}\right )^{2}}{x} \\ y \left (x \right ) &= \frac {\left (-x +\sqrt {c_{1} x}\right )^{2}}{x} \\ \end{align*}

Problem 4085

ODE	\[ \boxed {x {y^{\prime }}^{2}-2 y=-x} \]
program solution	\[ y = x \] Verified OK. \[ x = \frac {c_{2} {\mathrm e}^{\frac {2 x}{\sqrt {-x \left (x -2 y\right )}-x}} x^{2}}{\left (\sqrt {-x \left (x -2 y\right )}-x \right )^{2}} \] Verified OK. \[ x = \frac {c_{2} {\mathrm e}^{-\frac {2 x}{\sqrt {-x \left (x -2 y\right )}+x}} x^{2}}{\left (\sqrt {-x \left (x -2 y\right )}+x \right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right ) x}{2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}} \\ y \left (x \right ) &= \frac {\left (2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right ) x}{2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}} \\ \end{align*}

Problem 4086

ODE	\[ \boxed {x {y^{\prime }}^{2}+y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = x +1 \] Verified OK. \[ x = -\frac {2 \left (2 x \ln \left (2\right )-2 \ln \left (\frac {-1+\sqrt {1+4 y x}}{x}\right ) x -2 c_{1} x +\sqrt {1+4 y x}-1\right ) x}{\left (-1+\sqrt {1+4 y x}-2 x \right )^{2}} \] Verified OK. \[ x = \frac {2 \left (-2 x \ln \left (2\right )+2 \ln \left (\frac {-1-\sqrt {1+4 y x}}{x}\right ) x +2 c_{1} x +\sqrt {1+4 y x}+1\right ) x}{\left (1+\sqrt {1+4 y x}+2 x \right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +\textit {\_Z} +c_{1} -x -{\mathrm e}^{\textit {\_Z}}\right )} x +\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +\textit {\_Z} +c_{1} -x -{\mathrm e}^{\textit {\_Z}}\right )+c_{1} -x \]

Problem 4087

ODE	\[ \boxed {x {y^{\prime }}^{2}+2 y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = x +2 \] Verified OK. \[ x = \frac {\left (2 \ln \left (\frac {-1+\sqrt {y x +1}}{x}\right ) x +c_{1} x -2 \sqrt {y x +1}+2\right ) x}{\left (1-\sqrt {y x +1}+x \right )^{2}} \] Verified OK. \[ x = \frac {\left (2 \ln \left (\frac {-1-\sqrt {y x +1}}{x}\right ) x +c_{1} x +2 \sqrt {y x +1}+2\right ) x}{\left (1+\sqrt {y x +1}+x \right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x -2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +2 \textit {\_Z} -x \right )} x +2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x -2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +2 \textit {\_Z} -x \right )+c_{1} -x \]

Problem 4088

ODE	\[ \boxed {x {y^{\prime }}^{2}-2 y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = x -2 \] Verified OK. \[ x = \frac {\left (-2 \ln \left (\frac {1+\sqrt {y x +1}}{x}\right ) x +c_{1} x +2 \sqrt {y x +1}+2\right ) x}{\left (1+\sqrt {y x +1}-x \right )^{2}} \] Verified OK. \[ x = \frac {\left (-2 \ln \left (\frac {1-\sqrt {y x +1}}{x}\right ) x +c_{1} x -2 \sqrt {y x +1}+2\right ) x}{\left (-1+\sqrt {y x +1}+x \right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} -2 \textit {\_Z} -x \right )} x -2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} -2 \textit {\_Z} -x \right )+c_{1} -x \]

Problem 4089

ODE	\[ \boxed {x {y^{\prime }}^{2}+4 y^{\prime }-2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = 2 x +4 \] Verified OK. \[ x = \frac {\left (8 \ln \left (\frac {-2+\sqrt {4+2 y x}}{x}\right ) x +c_{1} x -4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2-\sqrt {4+2 y x}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 \ln \left (\frac {-2-\sqrt {4+2 y x}}{x}\right ) x +c_{1} x +4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2+\sqrt {4+2 y x}\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +8 \textit {\_Z} -4 x \right )} x +4 \operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +8 \textit {\_Z} -4 x \right )+\frac {c_{1}}{2}-2 x \]

Problem 4090

ODE	\[ \boxed {x {y^{\prime }}^{2}+x y^{\prime }-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{\frac {2 x}{-x +\sqrt {x \left (4 y+x \right )}}}}{\left (-x +\sqrt {x \left (4 y+x \right )}\right )^{2}} \] Verified OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{-\frac {2 x}{x +\sqrt {x \left (4 y+x \right )}}}}{\left (x +\sqrt {x \left (4 y+x \right )}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right ) x}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ y \left (x \right ) &= \frac {\left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right ) x}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ \end{align*}

Problem 4091

ODE	\[ \boxed {x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }=-x} \]
program solution	\[ y = \frac {x^{2}}{2}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ \end{align*}

Problem 4092

ODE	\[ \boxed {x {y^{\prime }}^{2}+y^{\prime } y=-a} \]
program solution	\[ x = -\frac {4 a \,x^{2}}{3 \left (-y+\sqrt {y^{2}-4 a x}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y+\sqrt {y^{2}-4 a x}}{x}}} \] Verified OK. \[ x = -\frac {4 a \,x^{2}}{3 \left (y+\sqrt {y^{2}-4 a x}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y-\sqrt {y^{2}-4 a x}}{x}}} \] Verified OK.
Maple solution	\begin{align*} -\frac {8 \left (-\frac {3 c_{1} \left (y \left (x \right )-\sqrt {-4 a x +y \left (x \right )^{2}}\right ) \sqrt {\frac {-y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}}{x}}}{8}+a x -\frac {3 y \left (x \right )^{2}}{4}+\frac {3 y \left (x \right ) \sqrt {-4 a x +y \left (x \right )^{2}}}{4}\right ) x}{3 \left (y \left (x \right )-\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}} &= 0 \\ -\frac {8 x \left (\frac {3 c_{1} \left (y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right ) \sqrt {\frac {-2 y \left (x \right )-2 \sqrt {-4 a x +y \left (x \right )^{2}}}{x}}}{4}+a x -\frac {3 y \left (x \right )^{2}}{4}-\frac {3 y \left (x \right ) \sqrt {-4 a x +y \left (x \right )^{2}}}{4}\right )}{3 \left (y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}} &= 0 \\ \end{align*}

Problem 4093

ODE	\[ \boxed {x {y^{\prime }}^{2}-y^{\prime } y=-a} \]
program solution	\[ y = c_{1} x +\frac {a}{c_{1}} \] Verified OK. \[ y = \frac {2 a x}{\sqrt {a x}} \] Verified OK. \[ y = -\frac {2 a x}{\sqrt {a x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -2 \sqrt {a x} \\ y \left (x \right ) &= 2 \sqrt {a x} \\ y \left (x \right ) &= \frac {x \,c_{1}^{2}+a}{c_{1}} \\ \end{align*}

Problem 4094

ODE	\[ \boxed {x {y^{\prime }}^{2}-y^{\prime } y=-a x} \]
program solution	\[ x = \frac {c_{2} \left (y+\sqrt {y^{2}-4 a \,x^{2}}\right ) {\mathrm e}^{-\frac {\left (y+\sqrt {y^{2}-4 a \,x^{2}}\right )^{2}}{8 a \,x^{2}}}}{2 x} \] Verified OK. \[ x = -\frac {c_{2} \left (-y+\sqrt {y^{2}-4 a \,x^{2}}\right ) {\mathrm e}^{-\frac {\left (-y+\sqrt {y^{2}-4 a \,x^{2}}\right )^{2}}{8 a \,x^{2}}}}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-\operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )+1\right ) a c_{1} \sqrt {-\frac {x^{2}}{c_{1}^{2} a \operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )}} \]

Problem 4095

ODE	\[ \boxed {x {y^{\prime }}^{2}+y^{\prime } y=-x^{3}} \]
program solution	\[ \frac {\ln \left (y+\sqrt {y^{2}-4 x^{4}}\right )}{12}-\frac {5 \ln \left (-5 \sqrt {6}\, \sqrt {y^{2}-4 x^{4}}-\sqrt {6}\, y-24 i x^{2}\right )}{24}+\frac {5 \ln \left (x^{2} \sqrt {6}+6 i y\right )}{24}-\frac {5 \ln \left (x^{2} \sqrt {6}-6 i y\right )}{24}+\frac {5 \ln \left (5 \sqrt {6}\, \sqrt {y^{2}-4 x^{4}}-\sqrt {6}\, y+24 i x^{2}\right )}{24}+\frac {5 \ln \left (x^{4}+6 y^{2}\right )}{24} = c_{1} \] Verified OK.
Maple solution	\begin{align*} -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\textit {\_a} \left (5 y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}\right )}d \textit {\_a} \right )-2 \left (\int _{}^{y \left (x \right )}\frac {1+\left (40 \textit {\_f} -8 \sqrt {-4 x^{4}+\textit {\_f}^{2}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{5 \textit {\_f} -\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}+5 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} \right )+2 \left (\int _{}^{y \left (x \right )}\frac {-1+8 \left (\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

Problem 4096

ODE	\[ \boxed {x {y^{\prime }}^{2}-y^{\prime } y+a y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {c_{2} \left (-2 a x +y+\sqrt {y^{2}-4 x a y}\right ) {\mathrm e}^{-\frac {y+\sqrt {y^{2}-4 x a y}}{2 x a}}}{2 x} \] Verified OK. \[ x = -\frac {c_{2} \left (2 a x -y+\sqrt {y^{2}-4 x a y}\right ) {\mathrm e}^{\frac {-y+\sqrt {y^{2}-4 x a y}}{2 x a}}}{2 x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_{1} a}\right )-1\right )^{2} a x}{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_{1} a}\right )} \\ \end{align*}

Problem 4097

ODE	\[ \boxed {x {y^{\prime }}^{2}+y^{\prime } y-y^{4}=0} \]
program solution	\[ -\operatorname {arctanh}\left (\frac {1}{\sqrt {1+4 x y^{2}}}\right ) = \frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK. \[ \operatorname {arctanh}\left (\frac {1}{\sqrt {1+4 x y^{2}}}\right ) = \frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{2 \sqrt {-x}} \\ y \left (x \right ) &= \frac {1}{2 \sqrt {-x}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{2 x} \\ y \left (x \right ) &= \frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{2 x} \\ \end{align*}

Problem 4098

ODE	\[ \boxed {x {y^{\prime }}^{2}+\left (a -y\right ) y^{\prime }=-b} \]
program solution	\[ y = c_{1} x +\frac {a c_{1} +b}{c_{1}} \] Verified OK. \[ y = \frac {2 b x +\sqrt {b x}\, a}{\sqrt {b x}} \] Verified OK. \[ y = \frac {\sqrt {b x}\, a -2 b x}{\sqrt {b x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= a -2 \sqrt {b x} \\ y \left (x \right ) &= a +2 \sqrt {b x} \\ y \left (x \right ) &= \frac {x \,c_{1}^{2}+c_{1} a +b}{c_{1}} \\ \end{align*}

Problem 4099

ODE	\[ \boxed {x {y^{\prime }}^{2}+\left (-y+x \right ) y^{\prime }-y=-1} \]
program solution	\[ y = c_{1} x +\frac {1}{c_{1} +1} \] Verified OK. \[ y = 2 \sqrt {x}-x \] Verified OK. \[ y = -x -2 \sqrt {x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -x -2 \sqrt {x} \\ y \left (x \right ) &= -x +2 \sqrt {x} \\ y \left (x \right ) &= \frac {x \,c_{1}^{2}+c_{1} x +1}{c_{1} +1} \\ \end{align*}

Problem 4100

ODE	\[ \boxed {x {y^{\prime }}^{2}+\left (a +x -y\right ) y^{\prime }-y=0} \]
program solution	\[ y = c_{1} x +\frac {c_{1} a}{c_{1} +1} \] Verified OK. \[ y = \frac {-x \sqrt {-a x}-2 a x +\sqrt {-a x}\, a}{\sqrt {-a x}} \] Verified OK. \[ y = \frac {\left (a -x \right ) \sqrt {-a x}+2 a x}{\sqrt {-a x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= a -x -2 \sqrt {-a x} \\ y \left (x \right ) &= a -x +2 \sqrt {-a x} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1} x +a +x \right )}{c_{1} +1} \\ \end{align*}