63.1.77 problem 122

Internal problem ID [15517]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 122
Date solved : Thursday, October 02, 2025 at 10:19:21 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} -y^{\prime }+x y^{\prime \prime }&=x^{2} {\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 17

ode:=x*diff(diff(y(x),x),x)-diff(y(x),x) = x^2*exp(x); 
ic:=[y(0) = -1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (x -1\right ) {\mathrm e}^{x}+\frac {c_1 \,x^{2}}{2} \]

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 54

ode=x*D[y[x],{x,2}]-D[y[x],x]==x^2*Exp[x]; 
ic={y[0]==-1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x (x-1)\\ y(x)&\to \int _1^x\left (c_1+e^{K[1]}\right ) K[1]dK[1]-\int _1^0\left (c_1+e^{K[1]}\right ) K[1]dK[1]-1 \end{align*}

✓ Sympy. Time used: 0.232 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = C_{2} x^{2} + x e^{x} - e^{x} \]