Internal
problem
ID
[9063]
Book
:
First
order
enumerated
odes
Section
:
section
3.
First
order
odes
solved
using
Laplace
method
Problem
number
:
6
Date
solved
:
Wednesday, March 05, 2025 at 07:18:53 AM
CAS
classification
:
[_separable]
Solve
Since no initial condition is explicitly given, then let
We will now apply Laplace transform to each term in the ode. Since this is time varying, the following Laplace transform property will be used
Where in the above \(F(s)\) is the laplace transform of \(f \left (t \right )\). Applying the above property to each term of the ode gives
Collecting all the terms above, the ode in Laplace domain becomes
The above ode in Y(s) is now solved.
Since the ode has the form \(Y^{\prime }=f(s)\), then we only need to integrate \(f(s)\).
Applying inverse Laplace transform on the above gives.
Substituting initial conditions \(y \left (0\right ) = c_1\) and \(y^{\prime }\left (0\right ) = c_1\) into the above solution Gives
Solving for the constant \(c_2\) from the above equation gives
Substituting the above back into the solution (1) gives
ode:=t*diff(y(t),t)+y(t) = 0; dsolve(ode,y(t),method='laplace');
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
Maple step by step
ode=t*D[y[t],t]+y[t]==0; ic={}; DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
from sympy import * t = symbols("t") y = Function("y") ode = Eq(t*Derivative(y(t), t) + y(t),0) ics = {} dsolve(ode,func=y(t),ics=ics)