77.13.3 problem 4

Internal problem ID [20464]
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 (E) at page 39
Problem number : 4
Date solved : Thursday, October 02, 2025 at 06:03:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 p y^{\prime }+\left (p^{2}+q^{2}\right ) y&={\mathrm e}^{k x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

ode:=diff(diff(y(x),x),x)+2*p*diff(y(x),x)+(p^2+q^2)*y(x) = exp(k*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-p x} \sin \left (q x \right ) c_2 +{\mathrm e}^{-p x} \cos \left (q x \right ) c_1 +\frac {{\mathrm e}^{k x}}{k^{2}+2 p k +p^{2}+q^{2}} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 61

ode=D[y[x],{x,2}]+2*p*D[y[x],x]+(p^2+q^2)*y[x]==Exp[k*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-(x (p+i q))} \left (\frac {e^{x (k+p+i q)}}{k^2+2 k p+p^2+q^2}+c_2 e^{2 i q x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
k = symbols("k") 
p = symbols("p") 
q = symbols("q") 
y = Function("y") 
ode = Eq(2*p*Derivative(y(x), x) + (p**2 + q**2)*y(x) - exp(k*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x \left (- p + i q\right )} + C_{2} e^{- x \left (p + i q\right )} + \frac {e^{k x}}{k^{2} + 2 k p + p^{2} + q^{2}} \]