15.11.32 problem 32

Internal problem ID [3142]
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 : 32
Date solved : Sunday, March 30, 2025 at 01:19:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=3 x \sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = 3*sin(x)*x; 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \sin \left (x \right )+2 \cos \left (x \right )-\frac {3 \cos \left (x \right ) x^{2}}{4}+\frac {3 x \sin \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 28

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

\[ y(x)\to \left (2-\frac {3 x^2}{4}\right ) \cos (x)+\left (\frac {3 x}{4}+1\right ) \sin (x) \]

✓ Sympy. Time used: 0.140 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*sin(x) + y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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