Internal
problem
ID
[9055]
Book
:
First
order
enumerated
odes
Section
:
section
2
(system
of
first
order
odes)
Problem
number
:
1
Date
solved
:
Friday, February 21, 2025 at 09:07:24 PM
CAS
classification
:
system_of_ODEs
In canonical form a linear first order is
Comparing the above to the given ode shows that
The integrating factor \(\mu \) is
The ode becomes
Integrating gives
Dividing throughout by the integrating factor \({\mathrm e}^{-t}\) gives the final solution
The system is
Since the left side is the same, this implies
Taking derivative of the above w.r.t. \(t\) gives
Substituting (3,4) in (1) to eliminate \(y,y^{\prime }\) gives
Which is now solved for \(x\). Given now that we have the solution
Then substituting (6) into (3) gives
Solving time : 0.030
(sec)
Leaf size : 30
dsolve([diff(x(t),t)+diff(y(t),t)-x(t) = y(t)+t, diff(x(t),t)+diff(y(t),t) = 2*x(t)+3*y(t)+exp(t)],{op([x(t), y(t)])})
Solving time : 0.045
(sec)
Leaf size : 37
DSolve[{{D[x[t],t]+D[y[t],t]-x[t]==y[t]+t,D[x[t],t]+D[y[t],t]==2*x[t]+3*y[t]+Exp[t]},{}},{x[t],y[t]},t,IncludeSingularSolutions->True]
Solving time : 0.041
(sec)
Leaf size : 0
Python version: 3.13.1 (main, Dec 4 2024, 18:05:56) [GCC 14.2.1 20240910] Sympy version 1.13.3
from sympy import * t = symbols("t") x = Function("x") y = Function("y") ode=[Eq(-t - x(t) - y(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-2*x(t) - 3*y(t) - exp(t) + Derivative(x(t), t) + Derivative(y(t), t),0)] ics = {} dsolve(ode,func=[x(t),y(t)],ics=ics)
[]