75.25.2 problem 758

Internal problem ID [17154]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.3. Finding periodic solutions of linear differential equations. Exercises page 187
Problem number : 758
Date solved : Monday, March 31, 2025 at 03:43:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = Pi^2-x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {3}{8}+\left (c_1 x +c_2 \right ) {\mathrm e}^{2 x}-\frac {x^{2}}{4}+\frac {\pi ^{2}}{4}-\frac {x}{2} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 42

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

\[ y(x)\to \frac {1}{8} \left (-2 x^2-4 x+2 \pi ^2-3\right )+c_1 e^{2 x}+c_2 e^{2 x} x \]

✓ Sympy. Time used: 0.209 (sec). Leaf size: 29

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)) - pi**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x^{2}}{4} - \frac {x}{2} + \left (C_{1} + C_{2} x\right ) e^{2 x} - \frac {3}{8} + \frac {\pi ^{2}}{4} \]