86.11.8 problem 7 (b)

Internal problem ID [23202]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 9. The operational method. Exercise 9c at page 137
Problem number : 7 (b)
Date solved : Thursday, October 02, 2025 at 09:24:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=x^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = x^2; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (2 x -3\right ) {\mathrm e}^{2 x}}{8}+\frac {x^{2}}{4}+\frac {x}{2}+\frac {3}{8} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-4*D[y[x],x]+4*y[x]==x^2; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (2 x^2+4 x+e^{2 x} (2 x-3)+3\right ) \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 4*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} + \left (\frac {x}{4} - \frac {3}{8}\right ) e^{2 x} + \frac {3}{8} \]