Differential Equations Examples

From UBC Wiki
Jump to: navigation, search
MathHelp.png This article is part of the MathHelp Tutoring Wiki


Initial Value Problem

Example 1 :
Question:

Solve the following initial value problem.
, y(0) = 5
Given that the general solution is
Solution:
Since we know y(0) = 5.
That means t = 0.
Therefore
The desired solution is


Example 2 :
Question:

Solve the following initial value problem.
, y(0) = 3
Given that the general solution is
Solution:
Since we know y(0) = 3.
That means t = 0.
Therefore
The desired solution is

First-Order Linear Differential Equations

To solve first-order differential equations, you'll need to know the following facts:
  • A first-order linear differential equation in standard form is y' + a(t)y = b(t).
  • Find an integrating factor by integrating e^(∫a(t)dt) which will become the integrating factor in the form .
  • After find the integrating factor, multiply the first-order linear differential equation by the integrating factor.
  • Then be able to recognize how to combine terms in y' and y into a single term after multiplying the differential equation by the integrating factor, for example:
  • After combining the terms into a single term, integrate both sides to solve for y.
  • By solving y, you can find the general solution for y.

Example 1 :
Question:

Solve .
Solution:
Divide both sides by to ensure the differential equation is in its standard form (y' + a(t)y = b(t)).
So the differential equation turns into
Find the integrating factor by integrating e^(∫a(t)dt) = e^(∫
Then the integrating factor is .
Multiply the differential equation by the integrating factor t.
Recognize that the left hand side has the terms y' and y and you can combine them into a single term.
After combining it into a single term, it is .
You'll probably notice that you can combine the terms by recognizing that the single term is y times integrating factor = and in this example, it is yt.
Now integrate both sides and you'll get ty = ∫.
Now isolate y and get .
The general solution is .


Example 2 :
Question:

Solve .
Solution:
We know that a(t) = 1 and b(t) = 1.
Integrating factor = e^(∫1dt) = .
The general solution is or .