87.17.19 problem 19

Internal problem ID [23611]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 127
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:43:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

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

\[ y = {\mathrm e}^{\frac {x}{2}} c_2 +{\mathrm e}^{-x} c_1 +\frac {\left (\frac {19}{2}+x^{2}-5 x \right ) {\mathrm e}^{x}}{2} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {1}{4} e^x \left (2 x^2-10 x+19\right )+c_1 e^{x/2}+c_2 e^{-x} \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{\frac {x}{2}} + \frac {\left (2 x^{2} - 10 x + 19\right ) e^{x}}{4} \]