82.33.6 problem Ex. 6

Internal problem ID [18899]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 6
Date solved : Tuesday, January 28, 2025 at 12:34:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-a^{2} y&={\mathrm e}^{a x}+{\mathrm e}^{n x} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 72

dsolve(diff(y(x),x$2)-a^2*y(x)=exp(a*x)+exp(n*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {c_{2} \left (a^{4}-a^{2} n^{2}\right ) {\mathrm e}^{-a x}+\left (c_{1} a^{2}+\frac {1}{2} a x -\frac {1}{4}\right ) \left (a -n \right ) \left (a +n \right ) {\mathrm e}^{a x}-a^{2} {\mathrm e}^{n x}}{a^{4}-a^{2} n^{2}} \]

✓ Solution by Mathematica

Time used: 0.128 (sec). Leaf size: 121

DSolve[D[y[x],{x,2}]-a^2*y[x]==Exp[a*x]+Exp[n*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-a x} \left (4 a^4 \left (c_1 e^{2 a x}+c_2\right )+2 a^3 x e^{2 a x}-a^2 \left (e^{2 a x} \left (1+4 c_1 n^2\right )+4 e^{x (a+n)}+4 c_2 n^2\right )-2 a n^2 x e^{2 a x}+n^2 e^{2 a x}\right )}{4 a^2 (a-n) (a+n)} \]