66.2.25 problem Problem 34

Internal problem ID [13924]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 34
Date solved : Tuesday, January 28, 2025 at 06:08:47 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 40

dsolve(diff(y(x),x$3)-y(x)=exp(x),y(x), singsol=all)
 

\[ y = c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {{\mathrm e}^{x} \left (3 c_{1} +x \right )}{3} \]

✓ Solution by Mathematica

Time used: 0.368 (sec). Leaf size: 62

DSolve[D[y[x],{x,3}]-y[x]==Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} e^{-x/2} \left (e^{3 x/2} (x-1+3 c_1)+3 c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+3 c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]