First Order Linear D.E. using Separation of Variables

Example: Solve the d.e. dy/dx + y sec² x = 0

Strategy: Use separation of variables. i.e. separate y with x.

Rewrite the d.e. as follows: dy/dx = ˗ y sec² x

Next separate variables as follows: dy/y = ˗ sec² x dx

Integrate both sides of the d.e. :

ln y = ˗ tan x + C₁ where C₁ is the constant of integration

Solve for y(x). y(x) = C e ^{˗ tanx}

Suppose the initial condition is y(0) = 1, then

In this case 1 = C giving y(x) = e ^–tanx (result)

Check:

One nice thing about d.e.’s is you can check your answer to verify that your solution, y(x),

satisfies both the d.e. and the initial condition, in this case, y(0) = 1, by taking the

derivative of y(x)

dy/dx = - sec² x e ^–tanx = - sec² x y(x)

or dy/dx + sec² x y(x) = 0 which is the original d.e.

and y(0) = 1 (Check)

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