75.16.47 problem 520

Internal problem ID [16949]
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 : 520
Date solved : Monday, March 31, 2025 at 03:36:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 28

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

\[ y(x)\to \frac {e^x}{(k-1)^2}+(c_2 x+c_1) e^{k x} \]

✓ Sympy. Time used: 0.232 (sec). Leaf size: 24

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

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