75.16.45 problem 518

Internal problem ID [16947]
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 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 518
Date solved : Monday, March 31, 2025 at 03:36:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.178 (sec). Leaf size: 24

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 = {} 
dsolve(ode,func=y(x),ics=ics)
 

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