Differential Equations Examples
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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 .
- Back to Integral Calculus
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