23.3.97 problem 99

Internal problem ID [5811]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 99
Date solved : Tuesday, September 30, 2025 at 02:03:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} b^{2} y+2 a y^{\prime }+y^{\prime \prime }&=c \sin \left (k x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 123

ode:=b^2*y(x)+2*a*diff(y(x),x)+diff(diff(y(x),x),x) = c*sin(k*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-2 \cos \left (k x \right ) a c k +c \left (b^{2}-k^{2}\right ) \sin \left (k x \right )+4 \left ({\mathrm e}^{\left (-a +\sqrt {a^{2}-b^{2}}\right ) x} c_2 +{\mathrm e}^{-\left (a +\sqrt {a^{2}-b^{2}}\right ) x} c_1 \right ) \left (a^{2} k^{2}+\frac {1}{4} b^{4}-\frac {1}{2} b^{2} k^{2}+\frac {1}{4} k^{4}\right )}{4 a^{2} k^{2}+b^{4}-2 b^{2} k^{2}+k^{4}} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 103

ode=b^2*y[x] + 2*a*D[y[x],x] + D[y[x],{x,2}] == c*Sin[k*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{-\left (x \left (\sqrt {a^2-b^2}+a\right )\right )}+c_2 e^{x \left (\sqrt {a^2-b^2}-a\right )}+\frac {c \left (\left (b^2-k^2\right ) \sin (k x)-2 a k \cos (k x)\right )}{4 a^2 k^2+b^4-2 b^2 k^2+k^4} \end{align*}

✓ Sympy. Time used: 0.256 (sec). Leaf size: 139

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

\[ y{\left (x \right )} = C_{1} e^{x \left (- a + \sqrt {a^{2} - b^{2}}\right )} + C_{2} e^{- x \left (a + \sqrt {a^{2} - b^{2}}\right )} - \frac {2 a c k \cos {\left (k x \right )}}{4 a^{2} k^{2} + b^{4} - 2 b^{2} k^{2} + k^{4}} + \frac {b^{2} c \sin {\left (k x \right )}}{4 a^{2} k^{2} + b^{4} - 2 b^{2} k^{2} + k^{4}} - \frac {c k^{2} \sin {\left (k x \right )}}{4 a^{2} k^{2} + b^{4} - 2 b^{2} k^{2} + k^{4}} \]