89.19.27 problem 27

Internal problem ID [24755]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 27
Date solved : Thursday, October 02, 2025 at 10:47:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

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

\[ y = {\mathrm e}^{a x} \left (\int \left (1+{\mathrm e}^{-a x} f^{\prime \prime }\left (x \right )\right )d x x +c_1 x -\int x \left (1+{\mathrm e}^{-a x} f^{\prime \prime }\left (x \right )\right )d x +c_2 \right ) \]

✓ Mathematica. Time used: 0.261 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to e^{a x} \left (\int _1^xK[1] \left (-e^{-a K[1]} f''(K[1])-1\right )dK[1]+x \int _1^x\left (e^{-a K[2]} f''(K[2])+1\right )dK[2]+c_2 x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.843 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x}{2} + \int e^{- a x} \frac {d^{2}}{d x^{2}} f{\left (x \right )}\, dx\right ) - \int x e^{- a x} \frac {d^{2}}{d x^{2}} f{\left (x \right )}\, dx\right ) e^{a x} \]