64.11.50 problem 50

Internal problem ID [13500]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 50
Date solved : Tuesday, January 28, 2025 at 05:49:25 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime }&=x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 91

dsolve(diff(y(x),x$6)+2*diff(y(x),x$5)+5*diff(y(x),x$4)=x^3+x^2*exp(-x)+exp(-x)*sin(2*x),y(x), singsol=all)
 

\[ y = x c_5 +c_6 +\frac {\left (\int \left (\left (\left (-330 x +1320 c_{1} +240 c_{2} +69\right ) \cos \left (2 x \right )+\left (60 x -240 c_{1} +1320 c_{2} +567\right ) \sin \left (2 x \right )-3750 x^{2}-22500 x -43125\right ) {\mathrm e}^{-x}+25 x^{6}-60 x^{5}-30 x^{4}+288 x^{3}+7500 x^{2} c_{3} +15000 c_4 x \right )d x \right )}{15000} \]

✓ Solution by Mathematica

Time used: 60.208 (sec). Leaf size: 161

DSolve[D[y[x],{x,6}]+2*D[y[x],{x,5}]+5*D[y[x],{x,4}]==x^3+x^2*Exp[-x]+Exp[-x]*Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\int _1^{K[6]}\int _1^{K[5]}\int _1^{K[4]}e^{-K[3]} \left (c_2 \cos (2 K[3])+\int _1^{K[3]}-\frac {1}{2} \sin (2 K[2]) \left (e^{K[2]} K[2]^3+K[2]^2+\sin (2 K[2])\right )dK[2] \cos (2 K[3])+c_1 \sin (2 K[3])+\sin (2 K[3]) \int _1^{K[3]}\frac {1}{2} \cos (2 K[1]) \left (e^{K[1]} K[1]^3+K[1]^2+\sin (2 K[1])\right )dK[1]\right )dK[3]dK[4]dK[5]dK[6]+x (x (c_6 x+c_5)+c_4)+c_3 \]