Solve the initial value problem 4 (sin (t) dy / dt + cos (t) y) = cos (t) sin^2 (t), for 0 less...


Solve the initial value problem {eq}\displaystyle 4 (\sin (t) \frac{dy}{dt} + \cos (t) y) = \cos (t) \sin^2 (t) {/eq}, for {eq}0 < t < \pi {/eq} and {eq}\displaystyle y \bigg(\frac{\pi}{2}\bigg) = 7 {/eq}.

Find the integrating factor, and find y (t).

Initial Value Problem:

The equation is of the type

{eq}\frac{\mathrm{d} y}{\mathrm{d} t}+Py=Q(f(t)) {/eq} where {eq}e^{Pdt} {/eq} is the integrating factor and the value of the constant can be found using the intial value conditions.

Answer and Explanation: 1

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To solve the equation let us write the general equation

{eq}\frac{\mathrm{d} y}{\mathrm{d} t}+Py=Q(f(t)) {/eq}

Let us rearrange the equation


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Learn more about this topic:

Integrating Factor: Method & Example


Chapter 12 / Lesson 6

Learn how to find integrating factors. Review the integrating factor method and formula to solve linear first- and second-order differential equations with examples.

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