90.20.33 problem 20
Internal
problem
ID
[25328]
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
337
Problem
number
:
20
Date
solved
:
Friday, October 03, 2025 at 12:00:12 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }-2 y^{\prime }-2 y&=\frac {t^{2}+1}{-t^{2}+1} \end{align*}
With initial conditions
\begin{align*}
y \left (2\right )&=y_{1} \\
y^{\prime }\left (2\right )&=y_{1} \\
\end{align*}
✓ Maple. Time used: 0.240 (sec). Leaf size: 253
ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)-2*y(t) = (t^2+1)/(-t^2+1);
ic:=[y(2) = y__1, D(y)(2) = y__1];
dsolve([ode,op(ic)],y(t), singsol=all);
\[
y = -\frac {{\mathrm e}^{-\left (\sqrt {3}-1\right ) \left (t -1\right )} \operatorname {Ei}_{1}\left (-\left (\sqrt {3}-1\right ) \left (t -1\right )\right ) \sqrt {3}}{6}+\frac {{\mathrm e}^{-\left (\sqrt {3}-1\right ) \left (t +1\right )} \operatorname {Ei}_{1}\left (-\left (\sqrt {3}-1\right ) \left (t +1\right )\right ) \sqrt {3}}{6}+\frac {\left (6 y_{1} -\sqrt {3}-3\right ) {\mathrm e}^{-\left (-2+t \right ) \left (\sqrt {3}-1\right )}}{12}+\frac {\sqrt {3}\, {\mathrm e}^{-\left (\sqrt {3}-1\right ) \left (t -1\right )} \operatorname {Ei}_{1}\left (-\sqrt {3}+1\right )}{6}-\frac {\sqrt {3}\, {\mathrm e}^{-\left (\sqrt {3}-1\right ) \left (t +1\right )} \operatorname {Ei}_{1}\left (-3 \sqrt {3}+3\right )}{6}+\frac {{\mathrm e}^{\left (1+\sqrt {3}\right ) \left (t -1\right )} \operatorname {Ei}_{1}\left (\left (1+\sqrt {3}\right ) \left (t -1\right )\right ) \sqrt {3}}{6}-\frac {{\mathrm e}^{\left (1+\sqrt {3}\right ) \left (t +1\right )} \operatorname {Ei}_{1}\left (\left (1+\sqrt {3}\right ) \left (t +1\right )\right ) \sqrt {3}}{6}+\frac {\left (6 y_{1} +\sqrt {3}-3\right ) {\mathrm e}^{\left (-2+t \right ) \left (1+\sqrt {3}\right )}}{12}+\frac {\sqrt {3}\, {\mathrm e}^{\left (1+\sqrt {3}\right ) \left (t +1\right )} \operatorname {Ei}_{1}\left (3+3 \sqrt {3}\right )}{6}-\frac {\sqrt {3}\, {\mathrm e}^{\left (1+\sqrt {3}\right ) \left (t -1\right )} \operatorname {Ei}_{1}\left (1+\sqrt {3}\right )}{6}+\frac {1}{2}
\]
✓ Mathematica. Time used: 0.668 (sec). Leaf size: 405
ode=D[y[t],{t,2}]-2*D[y[t],t]-2*y[t]==(1+t^2)/(1-t^2);
ic={y[2]==y1,Derivative[1][y][2] ==y1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {1}{12} e^{-\sqrt {3} t-2 \sqrt {3}-2} \left (2 \sqrt {3} e^{t+3 \sqrt {3}+1} \operatorname {ExpIntegralEi}\left (\left (-1+\sqrt {3}\right ) (t-1)\right )-2 \sqrt {3} e^{2 \sqrt {3} t+t+\sqrt {3}+1} \operatorname {ExpIntegralEi}\left (-\left (\left (1+\sqrt {3}\right ) (t-1)\right )\right )-2 \sqrt {3} e^{t+\sqrt {3}+3} \operatorname {ExpIntegralEi}\left (\left (-1+\sqrt {3}\right ) (t+1)\right )+2 \sqrt {3} e^{2 \sqrt {3} t+t+3 \sqrt {3}+3} \operatorname {ExpIntegralEi}\left (-\left (\left (1+\sqrt {3}\right ) (t+1)\right )\right )-2 \sqrt {3} \operatorname {ExpIntegralEi}\left (-3 \left (1+\sqrt {3}\right )\right ) e^{2 \sqrt {3} t+t+3 \sqrt {3}+3}+2 \sqrt {3} \operatorname {ExpIntegralEi}\left (3 \left (-1+\sqrt {3}\right )\right ) e^{t+\sqrt {3}+3}-2 \sqrt {3} \operatorname {ExpIntegralEi}\left (-1+\sqrt {3}\right ) e^{t+3 \sqrt {3}+1}+2 \sqrt {3} \operatorname {ExpIntegralEi}\left (-1-\sqrt {3}\right ) e^{2 \sqrt {3} t+t+\sqrt {3}+1}+6 e^{t+4 \sqrt {3}} \text {y1}+6 e^{2 \sqrt {3} t+t} \text {y1}-\sqrt {3} e^{t+4 \sqrt {3}}-3 e^{t+4 \sqrt {3}}+6 e^{\sqrt {3} t+2 \sqrt {3}+2}+\sqrt {3} e^{2 \sqrt {3} t+t}-3 e^{2 \sqrt {3} t+t}\right ) \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y1 = symbols("y1")
y = Function("y")
ode = Eq(-2*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - (t**2 + 1)/(1 - t**2),0)
ics = {y(2): y1, Subs(Derivative(y(t), t), t, 2): y1}
dsolve(ode,func=y(t),ics=ics)
NotImplementedError : The given ODE Derivative(y(t), t) - (-2*t**2*y(t) + t**2*Derivative(y(t), (t,