9.1.9-eg-1 - Example Solve the separable differential...

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Unformatted text preview: Example Solve the separable differential equation dy sin x − 4y cos2 x subject to the initial condition y π 3 dx = 0, = 1. First separate variables: dy =0 dx sin x − 4y cos2 x sin x dx, cos2 x 4y dy = then integrate both sides (using the substitution u = cos x): sin x dx cos2 x 4y dy = 2y 2 = − du u2 y2 = 1 +C 2 cos x y2 = 1 sec x + C. 2 Solving for the constant C using the initial condition y 1= 1 π sec +C 2 3 therefore y2 = ⇒ π 3 = 1 leads to 1 − 1 = C, 1 sec x. 2 Solving for y explicitly y=± 1 sec x 2 and using the initial condition, the solution is y ( x) = 1 sec x, 2 −π π <x< . 2 2 ...
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