89.5.23 problem 23

Internal problem ID [24373]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 43
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:22:30 PM
CAS classification : [_linear]

\begin{align*} \left (a^{2}+x^{2}\right ) y^{\prime }&=2 x \left (\left (a^{2}+x^{2}\right )^{2}+3 y\right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

ode:=(a^2+x^2)*diff(y(x),x) = 2*x*((a^2+x^2)^2+3*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (a^{2}+x^{2}\right )^{2} \left (-1+c_1 \left (a^{2}+x^{2}\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 27

ode=(x^2+a^2)*D[y[x],x]== 2*x*( (x^2+a^2)^2 + 3*y[x] ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (a^2+x^2\right )^2 \left (-1+c_1 \left (a^2+x^2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.479 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*x*((a**2 + x**2)**2 + 3*y(x)) + (a**2 + x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} a^{6} + 3 C_{1} a^{4} x^{2} + 3 C_{1} a^{2} x^{4} + C_{1} x^{6} - a^{4} - 2 a^{2} x^{2} - x^{4} \]