74.3.22 problem 17

Internal problem ID [15882]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 08:17:10 AM
CAS classification : [_linear]

\begin{align*} 2 y^{\prime }+t y&=\ln \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y \left ({\mathrm e}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.616 (sec). Leaf size: 97

dsolve([2*diff(y(t),t)+t*y(t)=ln(t),y(exp(1)) = 0],y(t), singsol=all)
 

\[ y = -\frac {\left (\operatorname {erfi}\left (\frac {{\mathrm e}}{2}\right ) \sqrt {\pi }\, \sqrt {-t^{2}}-\sqrt {\pi }\, \ln \left (t \right ) t \,\operatorname {erf}\left (\frac {\sqrt {-t^{2}}}{2}\right )+t \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {t^{2}}{4}\right ) \sqrt {-t^{2}}-{\mathrm e} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {{\mathrm e}^{2}}{4}\right ) \sqrt {-t^{2}}\right ) {\mathrm e}^{-\frac {t^{2}}{4}}}{2 \sqrt {-t^{2}}} \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 39

DSolve[{2*D[y[t],t]+t*y[t]==Log[t],{y[Exp[1]]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-\frac {t^2}{4}} \int _e^t\frac {1}{2} e^{\frac {K[1]^2}{4}} \log (K[1])dK[1] \]