15.15.16 problem 17

Internal problem ID [3220]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 24, page 109
Problem number : 17
Date solved : Monday, January 27, 2025 at 07:26:30 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{-x} \left (-\sin \left (x \right ) x^{2}-2 \cos \left (x \right ) x +\sin \left (x \right )\right )}{2}-\frac {{\mathrm e}^{-2 x} c_{1}}{2}+c_2 \]

✓ Solution by Mathematica

Time used: 1.016 (sec). Leaf size: 39

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

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