87.17.48 problem 57

Internal problem ID [23640]
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 : 57
Date solved : Thursday, October 02, 2025 at 09:43:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \\ y^{\prime }\left (0\right )&=y_{1} \\ \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 54

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = exp(a*x); 
ic:=[y(0) = y__0, D(y)(0) = y__1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{x} \left (-1+y_{0} \left (a -1\right )^{2}\right )}{\left (a -1\right )^{2}}+\frac {{\mathrm e}^{x} x \left (\left (-y_{0} +y_{1} \right ) a +y_{0} -y_{1} -1\right )}{a -1}+\frac {{\mathrm e}^{a x}}{\left (a -1\right )^{2}} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 51

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==Exp[a*x]; 
ic={y[0]==y0,Derivative[1][y][0] ==y1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^x \left (a^2 \text {y0}-(a-1) x ((a-1) \text {y0}-a \text {y1}+\text {y1}+1)+e^{(a-1) x}-2 a \text {y0}+\text {y0}-1\right )}{(a-1)^2} \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y0 = symbols("y0") 
y1 = symbols("y1") 
y = Function("y") 
ode = Eq(y(x) - exp(a*x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): y0, Subs(Derivative(y(x), x), x, 0): y1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {x \left (- a y_{0} + a y_{1} + y_{0} - y_{1} - 1\right )}{a - 1} + \frac {a^{2} y_{0} - 2 a y_{0} + y_{0} - 1}{a^{2} - 2 a + 1}\right ) e^{x} + \frac {e^{a x}}{a^{2} - 2 a + 1} \]