desolve_laplace

## Description

`desolve_laplace` will solve an ordinary differential equation using Laplace transforms with or without initial conditions. The following commands solve $\dfrac{dx}{dt} - x = e^{-t}$.

## Sage Cell

#### Code

```
t = var('t') # define a variable t
x = function('x')(t) # define x to be a function of that variable
DE = diff(x, t) - x == exp(-t)
desolve_laplace(DE, [x,t])
```

## Options

#### Option

Solving the initial-value problem $\dfrac{dx}{dt} - x = e^{-t}$, $x(0) = 2$.

#### Code

```
t = var('t') # define a variable t
x = function('x')(t) # define x to be a function of that variable
DE = diff(x, t) - x == exp(-t)
desolve_laplace(DE, [x,t], ics=[0,2])
```

## Tags

Primary Tags: Differential Equations: Laplace transforms

Secondary Tags: Laplace transforms: Solving initial-value problems

## Related Cells

- desolve. Solving ordinary differential equations with
`desolve`. - desolve_odeint. Solving ordinary differential equations numerically with
`desolve_odeint`. - Euler's Method.
`eulers_method`implements Euler’s method for finding a numerical solution of the first-order ODE $y′=f(x,y)$. - Euler's Method for Systems.
`eulers_method_2x2`implements Euler’s method for finding a numerical solution of a $2 \times 2$ system of first-order ODEs. - Laplace Transforms. Sage can compute Laplace transforms.
- Interact to plot direction fields and solutions for first order differential equations. A Sage interact for plotting direction fields for differential equations.

## Attribute

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Date: 06 Jan 2018 20:03

Submitted by: Tom Judson