75.14.4 problem 330

Internal problem ID [16839]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 330
Date solved : Thursday, March 13, 2025 at 08:52:58 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=x \,{\mathrm e}^{x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 15

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

\[ y(x)\to e^x (x-2)+x+2 \]

✓ Sympy. Time used: 0.078 (sec). Leaf size: 15

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

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