Internal
problem
ID
[9059]
Book
:
First
order
enumerated
odes
Section
:
section
3.
First
order
odes
solved
using
Laplace
method
Problem
number
:
2
Date
solved
:
Friday, February 21, 2025 at 09:07:27 PM
CAS
classification
:
[_separable]
Solve
With initial conditions
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
Replacing \(y \left (0\right ) = 0\) in the above results in
The above ode in Y(s) is now solved.
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 \({\mathrm e}^{\frac {s^{2}}{2}}\) gives the final solution
Applying inverse Laplace transform on the above gives.
Substituting initial conditions \(y \left (0\right ) = 0\) and \(y^{\prime }\left (0\right ) = 0\) into the above solution Gives
Solving for the constant \(c_1\) from the above equation gives
Substituting the above back into the solution (1) gives
| |
Solution \(y = 0\) | Slope field \(y^{\prime }-t y = 0\) |
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
Solving time : 1.427
(sec)
Leaf size : 5
dsolve([diff(y(t),t)-y(t)*t = 0,op([y(0) = 0])],y(t),method=laplace)
Solving time : 0.001
(sec)
Leaf size : 6
DSolve[{D[y[t],t]-t*y[t]==0,y[0]==0},y[t],t,IncludeSingularSolutions->True]
Solving time : 0.259
(sec)
Leaf size : 3
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") y = Function("y") ode = Eq(-t*y(t) + Derivative(y(t), t),0) ics = {y(0): 0} dsolve(ode,func=y(t),ics=ics)
Eq(y(t), 0)