63.1.40 problem 59

Internal problem ID [15480]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 59
Date solved : Thursday, October 02, 2025 at 10:18:19 AM
CAS classification : [_linear]

\begin{align*} \left (-x^{2}+x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y-a \,x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

ode:=(-x^2+x)*diff(y(x),x)+(2*x^2-1)*y(x)-a*x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\left ({\mathrm e}^{2 x -2} \operatorname {Ei}_{1}\left (2 x -2\right ) a \left (x -1\right )-c_1 \,{\mathrm e}^{2 x} \left (x -1\right )-a \right ) x \]

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 91

ode=(x-x^2)*D[y[x],x]+(2*x^2-1)*y[x]-a*x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {1-2 K[1]^2}{K[1]-K[1]^2}dK[1]\right ) \left (\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1-2 K[1]^2}{K[1]-K[1]^2}dK[1]\right ) K[2]^2}{1-K[2]}dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 6.844 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**3 + (-x**2 + x)*Derivative(y(x), x) + (2*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \frac {\left (a x^{3} - 2 x^{2} y{\left (x \right )} + y{\left (x \right )}\right ) e^{- 2 x}}{x^{2} \left (x - 1\right )^{2}}\, dx = C_{1} \]