15.15.14 problem 15

Internal problem ID [3218]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 24, page 109
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:21:32 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 33

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

\[ y = \frac {\left (\left (-15 x +17\right ) \cos \left (x \right )+\sin \left (x \right ) \left (5 x +6\right )\right ) {\mathrm e}^{2 x}}{50}+{\mathrm e}^{x} c_1 +c_2 \]

✓ Mathematica. Time used: 0.501 (sec). Leaf size: 46

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

\[ y(x)\to \frac {1}{50} e^{2 x} (5 x+6) \sin (x)-\frac {1}{50} e^{2 x} (15 x-17) \cos (x)+c_1 e^x+c_2 \]

✓ Sympy. Time used: 0.323 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} + C_{2} e^{x} + \frac {\left (5 x \sin {\left (x \right )} - 15 x \cos {\left (x \right )} + 6 \sin {\left (x \right )} + 17 \cos {\left (x \right )}\right ) e^{2 x}}{50} \]