76.33.24 problem Ex. 24

Internal problem ID [20202]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 24
Date solved : Friday, October 03, 2025 at 06:36:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=x \sin \left (x \right )+\left (x^{2}+1\right ) {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

ode:=diff(diff(y(x),x),x)-y(x) = x*sin(x)+(x^2+1)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} c_2 +\frac {\left (4 x^{3}-6 x^{2}+24 c_1 +18 x -9\right ) {\mathrm e}^{x}}{24}-\frac {x \sin \left (x \right )}{2}-\frac {\cos \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to \frac {1}{24} \left (e^x \left (4 x^3-6 x^2+18 x-9\right )-12 x \sin (x)-12 \cos (x)\right )+c_1 e^x+c_2 e^{-x} \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 39

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

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