40.15.2 problem 11
Internal
problem
ID
[6788]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
21.
System
of
simultaneous
linear
equations.
Supplemetary
problems.
Page
163
Problem
number
:
11
Date
solved
:
Sunday, March 30, 2025 at 11:22:51 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )&=t\\ 5 x \left (t \right )+\frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=t^{2} \end{align*}
✓ Maple. Time used: 0.156 (sec). Leaf size: 53
ode:=[diff(x(t),t)+2*x(t)+diff(y(t),t)+y(t) = t, 5*x(t)+diff(y(t),t)+3*y(t) = t^2];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 -t^{2}+t +3 \\
y \left (t \right ) &= 2 t^{2}+\frac {\cos \left (t \right ) c_2}{2}-\frac {\sin \left (t \right ) c_1}{2}-3 t -4-\frac {3 \sin \left (t \right ) c_2}{2}-\frac {3 \cos \left (t \right ) c_1}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.127 (sec). Leaf size: 61
ode={D[x[t],t]+2*x[t]+D[y[t],t]+y[t]==t,5*x[t]+D[y[t],t]+3*y[t]==t^2};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -t^2+t+c_1 \cos (t)+(3 c_1+2 c_2) \sin (t)+3 \\
y(t)\to 2 t^2-3 t+c_2 \cos (t)-(5 c_1+3 c_2) \sin (t)-4 \\
\end{align*}
✓ Sympy. Time used: 0.300 (sec). Leaf size: 133
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-t + 2*x(t) + y(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-t**2 + 5*x(t) + 3*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - t^{2} \sin ^{2}{\left (t \right )} - t^{2} \cos ^{2}{\left (t \right )} + t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )} + \left (\frac {C_{1}}{5} + \frac {3 C_{2}}{5}\right ) \sin {\left (t \right )} - \left (\frac {3 C_{1}}{5} - \frac {C_{2}}{5}\right ) \cos {\left (t \right )} + 3 \sin ^{2}{\left (t \right )} + 3 \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + 2 t^{2} \sin ^{2}{\left (t \right )} + 2 t^{2} \cos ^{2}{\left (t \right )} - 3 t \sin ^{2}{\left (t \right )} - 3 t \cos ^{2}{\left (t \right )} - 4 \sin ^{2}{\left (t \right )} - 4 \cos ^{2}{\left (t \right )}\right ]
\]