71.7.6 problem 6

Internal problem ID [14427]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.3, page 71
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 06:30:41 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (1\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.123 (sec). Leaf size: 36

dsolve([diff(y(x),x)=2*y(x)/x+exp(x),y(1) = 1/2],y(x), singsol=all)
 

\[ y = -\operatorname {Ei}_{1}\left (-x \right ) x^{2}+\operatorname {Ei}_{1}\left (-1\right ) x^{2}+\frac {\left (2 x \,{\mathrm e}+x -2 \,{\mathrm e}^{x}\right ) x}{2} \]

✓ Solution by Mathematica

Time used: 0.097 (sec). Leaf size: 32

DSolve[{D[y[x],x]==2*y[x]/x+Exp[x],{y[1]==1/2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x^2 \left (2 \int _1^x\frac {e^{K[1]}}{K[1]^2}dK[1]+1\right ) \]