23.1.645 problem 639

Internal problem ID [5252]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 639
Date solved : Tuesday, September 30, 2025 at 12:01:58 PM
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*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 1111

ode:=(x^2*a+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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.24 (sec). Leaf size: 744

ode=(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; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
y = Function("y") 
ode = Eq(2*a*x*y(x) + b*y(x)**2 + k*x**2 + (a*x**2 + 2*b*x*y(x) + c*y(x)**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out