29.23.6 problem 637

Internal problem ID [5228]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 637
Date solved : Monday, January 27, 2025 at 10:24:25 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 1107

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

✓ Solution by Mathematica

Time used: 60.368 (sec). Leaf size: 744

DSolve[(a x^2+2 b x y[x]+c y[x]^2)D[y[x],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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\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 (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-36 b x}{36 c} \\ \end{align*}