90.23.8 problem 12
Internal
problem
ID
[25361]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
5.
Second
Order
Linear
Differential
Equations.
Exercises
at
page
365
Problem
number
:
12
Date
solved
:
Friday, October 03, 2025 at 12:00:37 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} -t y^{\prime \prime }-2 y^{\prime }+y t&=0 \end{align*}
Using Laplace method
✓ Maple. Time used: 0.044 (sec). Leaf size: 37
ode:=-t*diff(diff(y(t),t),t)-2*diff(y(t),t)+t*y(t) = 0;
dsolve(ode,y(t),method='laplace');
\[
y = -\frac {y \left (0\right ) \delta \left (t \right ) \operatorname {Typesetting}\mcoloneq \operatorname {msup}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mi}\left (\text {``$\mathcal \{L\}$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mrow}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mo}\left (\text {``$-$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mn}\left (``1''\right )\right ), \operatorname {Typesetting}\mcoloneq \operatorname {msemantics}=\text {``atomic''}\right )\left (\operatorname {arctanh}\left (\textit {\_s1} \right ), \textit {\_s1} , 0\right )}{\delta \left (0\right )}+y \left (0\right ) \operatorname {Typesetting}\mcoloneq \operatorname {msup}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mi}\left (\text {``$\mathcal \{L\}$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mrow}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mo}\left (\text {``$-$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mn}\left (``1''\right )\right ), \operatorname {Typesetting}\mcoloneq \operatorname {msemantics}=\text {``atomic''}\right )\left (\operatorname {arctanh}\left (\textit {\_s1} \right ), \textit {\_s1} , t\right )+\frac {y \left (0\right ) \delta \left (t \right )}{\delta \left (0\right )}
\]
✓ Mathematica. Time used: 0.018 (sec). Leaf size: 28
ode=-t*D[y[t],{t,2}]-2*D[y[t],t]+t*y[t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {2 c_1 e^{-t}+c_2 e^t}{2 t} \end{align*}
✓ Sympy. Time used: 0.129 (sec). Leaf size: 24
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(t*y(t) - t*Derivative(y(t), (t, 2)) - 2*Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (i t\right ) + C_{2} Y_{\frac {1}{2}}\left (i t\right )}{\sqrt {t}}
\]