23.6 problem 637

Internal problem ID [3376]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 23
Problem number: 637.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]

Solve \begin {gather*} \boxed {\left (y^{2} c +2 b y x +a \,x^{2}\right ) y^{\prime }+k \,x^{2}+2 a x y+b y^{2}=0} \end {gather*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 1666

dsolve((a*x^2+2*b*x*y(x)+c*y(x)^2)*diff(y(x),x)+k*x^2+2*a*x*y(x)+b*y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \frac {\frac {\left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}-\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{c \left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{c}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{4 c}+\frac {c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{c \left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{c}-\frac {i \sqrt {3}\, \left (\frac {\left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}+\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{c \left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{4 c}+\frac {c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{c \left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{c}+\frac {i \sqrt {3}\, \left (\frac {\left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}+\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{c \left (12 a \,x^{3} c_{1}^{3} b c -8 b^{3} x^{3} c_{1}^{3}-4 c_{1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 a^{3} c c_{1}^{6} x^{6}-3 a^{2} b^{2} c_{1}^{6} x^{6}-6 a b c c_{1}^{6} k \,x^{6}+4 b^{3} c_{1}^{6} k \,x^{6}+c^{2} c_{1}^{6} k^{2} x^{6}+6 a \,x^{3} c_{1}^{3} b c -4 b^{3} x^{3} c_{1}^{3}-2 c_{1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.595 (sec). Leaf size: 1327

DSolve[(a x^2+2 b x y[x]+c y[x]^2)y'[x]+k x^2+2 a x y[x]+b y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2^{2/3} \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}+\frac {2 \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}}-2 b x}{2 c} \\ y(x)\to \frac {9 i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}+\frac {36 \sqrt [3]{-2} x^2 \left (a c-b^2\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}}-36 b x}{36 c} \\ y(x)\to \frac {-9\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}+\frac {36 (-1)^{2/3} \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (-3 a b c+2 b^3+c^2 k\right )+c^2 e^{3 c_1}\right ){}^2}+3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )}}-36 b x}{36 c} \\ y(x)\to \frac {-2 b x \sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}+\left (2 \sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-2 x^3 \left (-3 a b c+2 b^3+c^2 k\right )\right )^{2/3}+2 \sqrt [3]{2} x^2 \left (b^2-a c\right )}{2 c \sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}} \\ y(x)\to \frac {-2 \sqrt [3]{-1} 2^{2/3} \sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}+\frac {4 (-1)^{2/3} \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}}-4 b x}{4 c} \\ y(x)\to \frac {(-2)^{2/3} \sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}-\frac {2 \sqrt [3]{-2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {c^2 x^6 \left (4 a^3 c-3 a^2 b^2-6 a b c k+4 b^3 k+c^2 k^2\right )}-x^3 \left (-3 a b c+2 b^3+c^2 k\right )}}-2 b x}{2 c} \\ \end{align*}