First Order Linear Equations

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Linear Differential Equations ....

Properties:

* .... know what they are right off hand.

* .... have method that we know to solve these equations.

First order linear equations have the form y' + a(x)y(x) = b(x) as shown in the illustration below. There must not be any powers of the variable nor can the variable be an argument of any of the terms.(S.G. Krantz, 2005)

In the example shown below we are given the equation y' + (3x^2)*y = x^2. Both sides are then multiplied by an exponential factor wherein a(x) represents 3x^2. (in this case) This is done because the original equation is itself not separable. What results, however, sets us up for the use of the product rule!! (bottom of the page)

Having used the product rule (albeit in reverse, kinda) we integrate both sides yielding an equation that can be solved for y. Notice that it has one constant given that it was a first order equation.

You can go to DE Ex2 here...D.E. Ex2.

You can go to DE Ex1 here ...D.E. Ex1.