23.3.98 problem 100

Internal problem ID [5812]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 100
Date solved : Tuesday, September 30, 2025 at 02:03:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a^{2} y-2 a y^{\prime }+y^{\prime \prime }&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

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

\[ y = \frac {\left (a -1\right )^{2} \left (c_1 x +c_2 \right ) {\mathrm e}^{a x}+{\mathrm e}^{x}}{\left (a -1\right )^{2}} \]

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to \frac {e^x}{(a-1)^2}+e^{a x} (c_2 x+c_1) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 24

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

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