Question #f913b

1 Answer
Oct 7, 2017

A simple first ordinary differential equation: y(t)'+y(t)=3, the analytical solution reads: y(t)=(y01)et+1

Explanation:

Introduction
Differential equations can be classified essentially on: ordinary and partial. Furthermore, they can be classified into: linear and nonlinear.

Fundamentaly. linear differential equations do not have variables in their coefficients. The simplest example are ordinary differential equations (ODEs).

The general form for linear first order ODEs:

a(t)y(t)'+b(t)y(t)=g(t)

They all have analytical solutions.

A simple Linear ordinary differential equations

y(t)'+y(t)=3

Applying the following strategy, we can obtain the solution for any equation of this shape.

Multiply both side by μ(t), where μ(t) is a generic function:

μ(t)y(t)'+μ(t)y(t)=3μ(t)

See that it takes us to conclude:

μ(t)'=μ(t)

By basic calculus, we can find:

μ(t)=Cet, take C=1

Moreover, we can conclude that:

(ety(t))'=3et, take C=1

By calculus, we can find the solution:

y(t)=(y01)et+1

Some words about the solution

It does not matter the initial solution, it will always converge to it; it is a nice property since as long as you give enough time, the system will always come back to the initial state.

limty(t)=y0

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