77.14.6 problem 6

Internal problem ID [20470]
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 : 6
Date solved : Thursday, October 02, 2025 at 06:03:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\sin \left (3 x \right )-\cos \left (\frac {x}{2}\right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+y(x) = sin(3*x)-cos(1/2*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-4 \cos \left (x \right )^{2}+8 c_2 -2 x +1\right ) \sin \left (x \right )}{8}-\frac {1}{2}+\frac {\left (8 c_1 -1\right ) \cos \left (x \right )}{8} \]

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to \frac {1}{8} (-\sin (3 x)+(-2+8 c_1) \cos (x)-2 (x+1-4 c_2) \sin (x)-4) \end{align*}

✓ Sympy. Time used: 0.981 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{2} \cos {\left (x \right )} + \left (C_{1} - \frac {x}{4}\right ) \sin {\left (x \right )} - \frac {\sin {\left (3 x \right )}}{8} - \frac {1}{2} \]