15.11.34 problem 34

Internal problem ID [3144]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 34
Date solved : Monday, January 27, 2025 at 07:23:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 8 y^{\prime \prime }-y&=x \,{\mathrm e}^{-\frac {x}{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 45

dsolve([8*diff(y(x),x$2)-y(x)=x*exp(-x/2),y(0) = 3, D(y)(0) = 5],y(x), singsol=all)
 

\[ y = \frac {\left (-5-16 \sqrt {2}\right ) {\mathrm e}^{-\frac {\sqrt {2}\, x}{4}}}{2}+\frac {\left (-5+16 \sqrt {2}\right ) {\mathrm e}^{\frac {\sqrt {2}\, x}{4}}}{2}+\left (x +8\right ) {\mathrm e}^{-\frac {x}{2}} \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 83

DSolve[{8*D[y[x],{x,2}]-y[x]==x*Exp[-x/2],{y[0]==3,Derivative[1][y][0] ==5}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-\frac {1}{4} \left (2+\sqrt {2}\right ) x} \left (2 e^{\frac {x}{2 \sqrt {2}}} (x+8)-\left (5+16 \sqrt {2}\right ) e^{x/2}+\left (16 \sqrt {2}-5\right ) e^{\frac {1}{2} \left (1+\sqrt {2}\right ) x}\right ) \]