60.1.289 problem 290

Internal problem ID [10303]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 290
Date solved : Monday, January 27, 2025 at 06:52:00 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 1113

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

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

✓ Solution by Mathematica

Time used: 60.388 (sec). Leaf size: 744

DSolve[(a*y[x]^2+2*b*x*y[x]+c*x^2)*D[y[x],x]+b*y[x]^2+2*c*x*y[x]+d*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 (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}+\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 (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}}-2 b x}{2 a} \\ 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 (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}+\frac {18 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2 \left (a c-b^2\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}}-36 b x}{36 a} \\ 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 (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}+\frac {18 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3\right ){}^2}-a^2 d x^3+a^2 e^{3 c_1}+3 a b c x^3-2 b^3 x^3}}-36 b x}{36 a} \\ \end{align*}