62.33.4 problem Ex 4

Internal problem ID [12920]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 57. Dependent variable absent. Page 132
Problem number : Ex 4
Date solved : Monday, March 31, 2025 at 07:24:37 AM
CAS classification : [[_2nd_order, _quadrature]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x) = x*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x -2\right ) {\mathrm e}^{x}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 19

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

\[ y(x)\to e^x (x-2)+c_2 x+c_1 \]

✓ Sympy. Time used: 0.068 (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 = {} 
dsolve(ode,func=y(x),ics=ics)
 

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