29.13 problem Ex 15

Internal problem ID [10274]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number: Ex 15.
ODE order: 3.
ODE degree: 1.

CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-y-x \,{\mathrm e}^{x}-\left (\cos ^{2}\relax (x )\right )=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 121

dsolve(diff(y(x),x$3)-y(x)=x*exp(x)+cos(x)^2,y(x), singsol=all)
 

\[ y \relax (x ) = \frac {\cos \left (2 x \right )}{10 \left (5+2 \sqrt {3}\right ) \left (-5+2 \sqrt {3}\right )}+\frac {4 \sin \left (2 x \right )}{5 \left (5+2 \sqrt {3}\right ) \left (-5+2 \sqrt {3}\right )}-\frac {13 \left (3 x^{2} {\mathrm e}^{x}-6 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x}-9\right )}{18 \left (5+2 \sqrt {3}\right ) \left (-5+2 \sqrt {3}\right )}+c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \]

Solution by Mathematica

Time used: 2.182 (sec). Leaf size: 80

DSolve[y'''[x]-y[x]==x*Exp[x]+Cos[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{18} e^x (3 (x-2) x+4+18 c_1)+\frac {1}{130} (-8 \sin (2 x)-\cos (2 x)-65)+e^{-x/2} \left (c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \\ \end{align*}