2.10.1 Problem (a)
Internal
problem
ID
[20992]
Book
:
Ordinary
Differential
Equations.
By
Wolfgang
Walter.
Graduate
texts
in
Mathematics.
Springer.
NY.
QA372.W224
1998
Section
:
Chapter
IV.
Linear
Differential
Equations.
Excercises
V
at
page
173
Problem
number
:
(a)
Date
solved
:
Saturday, November 29, 2025 at 01:33:51 AM
CAS
classification
:
system_of_ODEs
\begin{align*}
x^{\prime }&=\left (3 t -1\right ) x-\left (1-t \right ) y+t \,{\mathrm e}^{t^{2}} \\
y^{\prime }&=-\left (t +2\right ) x+\left (t -2\right ) y-{\mathrm e}^{t^{2}} \\
\end{align*}
Does not currently support non autonomous system of first order linear differential
equations.
✓ Maple. Time used: 0.145 (sec). Leaf size: 115
ode:=[diff(x(t),t) = (3*t-1)*x(t)-(-t+1)*y(t)+t*exp(t^2), diff(y(t),t) = -(t+2)*x(t)+(t-2)*y(t)-exp(t^2)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \left (c_2 +\frac {1}{9} t^{3}+\frac {1}{9} t^{2}+\frac {4}{9} t +\frac {4}{9}\right ) {\mathrm e}^{t^{2}}+\left (3 \,{\mathrm e}^{t^{2}-3 t} t -2 \,{\mathrm e}^{t^{2}-3 t}\right ) c_1 \\
y \left (t \right ) &= -\frac {{\mathrm e}^{t^{2}} t^{3}}{9}-\frac {t^{2} {\mathrm e}^{t^{2}}}{9}-3 \,{\mathrm e}^{t^{2}-3 t} c_1 t -{\mathrm e}^{t^{2}} c_2 -\frac {t \,{\mathrm e}^{t^{2}}}{9}-7 \,{\mathrm e}^{t^{2}-3 t} c_1 -\frac {8 \,{\mathrm e}^{t^{2}}}{9} \\
\end{align*}
✓ Mathematica. Time used: 0.475 (sec). Leaf size: 196
ode={D[x[t],t]==(3*t-1)*x[t]-(1-t)*y[t]+t*Exp[t^2], D[y[t],t]==-(t+2)*x[t]+(t-2)*y[t]-Exp[t^2]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {c_1 e^{t^2-\frac {1}{2}} \sqrt {t-1}}{\sqrt {2-2 t}}+\frac {\sqrt {2} c_2 e^{t^2-3 t+\frac {5}{2}} \sqrt {t-1} (3 t-2)}{9 \sqrt {1-t}}+\frac {1}{81} e^{t^2} \left (9 t^3+9 t^2+36 t-8\right )\\ y(t)&\to \frac {1}{162} e^{t^2} \left (-2 \left (9 t^3+9 t^2+9 t+28\right )-\frac {81 \sqrt {\frac {2}{e}} c_1 (t-1)}{\sqrt {-(t-1)^2}}-\frac {18 \sqrt {2} c_2 e^{\frac {5}{2}-3 t} (t-1) (3 t+7)}{\sqrt {-(t-1)^2}}\right ) \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-t*exp(t**2) + (1 - t)*y(t) - (3*t - 1)*x(t) + Derivative(x(t), t),0),Eq((2 - t)*y(t) + (t + 2)*x(t) + exp(t**2) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
NotImplementedError :