15.11.30 problem 30

Internal problem ID [3140]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 30
Date solved : Sunday, March 30, 2025 at 01:19:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = exp(x)*sin(x); 
ic:=y(0) = 3, D(y)(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (-2 \,{\mathrm e}^{x}+17\right ) \cos \left (x \right )}{5}+\frac {\sin \left (x \right ) \left ({\mathrm e}^{x}+11\right )}{5} \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 28

ode=D[y[x],{x,2}]+y[x]==Exp[x]*Sin[x]; 
ic={y[0]==3,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{5} \left (\left (e^x+11\right ) \sin (x)+\left (17-2 e^x\right ) \cos (x)\right ) \]

✓ Sympy. Time used: 0.128 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - exp(x)*sin(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {17}{5} - \frac {2 e^{x}}{5}\right ) \cos {\left (x \right )} + \left (\frac {e^{x}}{5} + \frac {11}{5}\right ) \sin {\left (x \right )} \]