83.17.13 problem 13

Internal problem ID [19138]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter III. Ordinary linear differential equations with constant coefficients. Misc. Examples on chapter III at page 50
Problem number : 13
Date solved : Monday, March 31, 2025 at 06:49:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+10 y+37 \sin \left (3 x \right )&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.064 (sec). Leaf size: 46

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+10*y(x)+37*sin(3*x) = 0; 
ic:=y(1/2*Pi) = 3, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (-6 \cos \left (3 x \right )-2 \sin \left (3 x \right )\right ) {\mathrm e}^{-x +\frac {\pi }{2}}+\left (-3 \,{\mathrm e}^{-x}+6\right ) \cos \left (3 x \right )-\sin \left (3 x \right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 52

ode=D[y[x],{x,2}]+2*D[y[x],x]+10*y[x]+37*Sin[3*x]==0; 
ic={y[Pi/2]==3,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \left (\left (6 e^x-3-6 e^{\pi /2}\right ) \cos (3 x)-\left (e^x+2 e^{\pi /2}\right ) \sin (3 x)\right ) \]

✓ Sympy. Time used: 0.247 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(10*y(x) + 37*sin(3*x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/2): 3, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- 2 e^{\frac {\pi }{2}} \sin {\left (3 x \right )} + \left (- 6 e^{\frac {\pi }{2}} - 3\right ) \cos {\left (3 x \right )}\right ) e^{- x} - \sin {\left (3 x \right )} + 6 \cos {\left (3 x \right )} \]