62.29.13 problem Ex 15

Internal problem ID [12964]
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
Date solved : Tuesday, January 28, 2025 at 04:45:33 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y&=x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 61

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

\[ y = -\frac {1}{2}+c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}+\frac {\left (3 x^{2}+18 c_{1} -6 x +4\right ) {\mathrm e}^{x}}{18} \]

✓ Solution by Mathematica

Time used: 2.002 (sec). Leaf size: 260

DSolve[D[y[x],{x,3}]-y[x]==x*Exp[x]+Cos[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x/2} \left (e^{3 x/2} \int _1^x\frac {1}{6} e^{-K[1]} \left (\cos (2 K[1])+2 e^{K[1]} K[1]+1\right )dK[1]+\cos \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{\frac {K[2]}{2}} \left (\cos (2 K[2])+2 e^{K[2]} K[2]+1\right ) \left (\sqrt {3} \cos \left (\frac {1}{2} \sqrt {3} K[2]\right )-3 \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )}{6 \sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{\frac {K[3]}{2}} \left (\cos (2 K[3])+2 e^{K[3]} K[3]+1\right ) \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[3]\right )+\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} K[3]\right )\right )}{6 \sqrt {3}}dK[3]+c_1 e^{3 x/2}+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]