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
FirstOrder Linear Differential Equations
 To solve firstorder differential equations, you'll need to know the following facts:
 A firstorder 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 firstorder 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 .
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