87.17.30 problem 31

Internal problem ID [23622]
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 : 31
Date solved : Thursday, October 02, 2025 at 09:43:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+4 y&={\mathrm e}^{2 x} \cos \left (x \right )+{\mathrm e}^{2 x} \sin \left (x \right ) \end{align*}

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

ode:=diff(diff(y(x),x),x)-5*diff(y(x),x)+4*y(x) = exp(2*x)*cos(x)+exp(2*x)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{4 x} c_2 +{\mathrm e}^{x} c_1 -\frac {{\mathrm e}^{2 x} \left (\cos \left (x \right )+2 \sin \left (x \right )\right )}{5} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-5*D[y[x],{x,1}]+4*y[x]==Exp[2*x]*Cos[x]+Exp[2*x]*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^x+c_2 e^{4 x}-\frac {1}{5} e^{2 x} (2 \sin (x)+\cos (x)) \end{align*}

✓ Sympy. Time used: 0.258 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - exp(2*x)*sin(x) - exp(2*x)*cos(x) - 5*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{3 x} + \frac {\sqrt {2} \left (- 3 \sin {\left (x + \frac {\pi }{4} \right )} + \cos {\left (x + \frac {\pi }{4} \right )}\right ) e^{x}}{10}\right ) e^{x} \]