77.14.5 problem 5

Internal problem ID [20469]
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. Exercise III (F) at page 42
Problem number : 5
Date solved : Thursday, October 02, 2025 at 06:03:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=2 \sin \left (\frac {x}{2}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)-4*y(x) = 2*sin(1/2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x} c_2 +{\mathrm e}^{-2 x} c_1 -\frac {8 \sin \left (\frac {x}{2}\right )}{17} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to -\frac {8}{17} \sin \left (\frac {x}{2}\right )+c_1 e^{2 x}+c_2 e^{-2 x} \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{2 x} - \frac {8 \sin {\left (\frac {x}{2} \right )}}{17} \]