83.13.3 problem 4

Internal problem ID [19179]
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 : Tuesday, January 28, 2025 at 01:11:23 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 59

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

\[ y \left (x \right ) = \frac {\left (k^{2}+2 p k +p^{2}+q^{2}\right ) \left (c_{1} \cos \left (q x \right )+c_{2} \sin \left (q x \right )\right ) {\mathrm e}^{-p x}+{\mathrm e}^{k x}}{k^{2}+2 p k +p^{2}+q^{2}} \]

✓ Solution by Mathematica

Time used: 0.117 (sec). Leaf size: 61

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

\[ y(x)\to e^{-(x (p+i q))} \left (\frac {e^{x (k+p+i q)}}{k^2+2 k p+p^2+q^2}+c_2 e^{2 i q x}+c_1\right ) \]