74.3.22 problem 17
Internal
problem
ID
[15803]
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, March 13, 2025 at 06:44:52 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*}
✓ Maple. Time used: 0.616 (sec). Leaf size: 97
ode:=2*diff(y(t),t)+t*y(t) = ln(t);
ic:=y(exp(1)) = 0;
dsolve([ode,ic],y(t), singsol=all);
\[
y = -\frac {\left (\operatorname {erfi}\left (\frac {{\mathrm e}}{2}\right ) \sqrt {\pi }\, \sqrt {-t^{2}}-\sqrt {\pi }\, \ln \left (t \right ) t \,\operatorname {erf}\left (\frac {\sqrt {-t^{2}}}{2}\right )+t \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {t^{2}}{4}\right ) \sqrt {-t^{2}}-{\mathrm e} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {{\mathrm e}^{2}}{4}\right ) \sqrt {-t^{2}}\right ) {\mathrm e}^{-\frac {t^{2}}{4}}}{2 \sqrt {-t^{2}}}
\]
✓ Mathematica. Time used: 0.057 (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]
\[
y(t)\to e^{-\frac {t^2}{4}} \int _e^t\frac {1}{2} e^{\frac {K[1]^2}{4}} \log (K[1])dK[1]
\]
✓ Sympy. Time used: 3.073 (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)
\[
y{\left (t \right )} = \left (- \frac {t {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {t^{2}}{4}} \right )}}{2} + \frac {\sqrt {\pi } \log {\left (t \right )} \operatorname {erfi}{\left (\frac {t}{2} \right )}}{2} + \frac {e {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {e^{2}}{4}} \right )}}{2} - \frac {\sqrt {\pi } \operatorname {erfi}{\left (\frac {e}{2} \right )}}{2}\right ) e^{- \frac {t^{2}}{4}}
\]