68.3.22 problem 17

Internal problem ID [17169]
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 : Thursday, October 02, 2025 at 01:49:02 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 28

ode:=2*diff(y(t),t)+t*y(t) = ln(t); 
ic:=[y(exp(1)) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\int _{{\mathrm e}}^{t}\ln \left (\textit {\_z1} \right ) {\mathrm e}^{\frac {\textit {\_z1}^{2}}{4}}d \textit {\_z1} {\mathrm e}^{-\frac {t^{2}}{4}}}{2} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 39

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

\begin{align*} 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] \end{align*}

✓ Sympy. Time used: 1.943 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*y(t) - log(t) + 2*Derivative(y(t), t),0) 
ics = {y(E): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {t {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {t^{2}}{4}} \right )}}{2} + \frac {\sqrt {\pi } \log {\left (t \right )} \operatorname {erfi}{\left (\frac {t}{2} \right )}}{2} + \frac {e {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {e^{2}}{4}} \right )}}{2} - \frac {\sqrt {\pi } \operatorname {erfi}{\left (\frac {e}{2} \right )}}{2}\right ) e^{- \frac {t^{2}}{4}} \]