If (dy)/(dx)+2 y tanx=sinx and y((π)/(3))=0, then the maximum value of y is equal to (a) (1)/(8)

step 1 We have to find the maximum value of y and we have given the differential equation that is DY by

If (dy)/(dx)+2 y tanx=sinx and y((π)/(3))=0, then the...

If $\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y} \tan \mathrm{x}=\sin \mathrm{x}$ and $\mathrm{y}\left(\frac{\pi}{3}\right)=0$, then the maximum value of $\mathrm{y}$ is equal to
(a) $\frac{1}{8}$
(b) $\frac{1}{4}$
(c) $\frac{1}{2}$
(d) 2

Key Concepts

First Order Linear Differential Equations

These are differential equations of the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are known functions. They frequently arise in various applications and their solution methodology is well-established, often involving the integrating factor method to obtain an explicit solution for the dependent variable.

Integrating Factor Method

This method is an essential technique for solving first order linear differential equations. By multiplying the entire differential equation by a strategically chosen function called the integrating factor, the left-hand side becomes the derivative of a product involving the unknown function, which then allows straightforward integration to find the solution.

Initial Value Problem (IVP)

An initial value problem specifies not only a differential equation but also the value of the unknown function at a particular point. This additional information is crucial as it allows one to determine the constant of integration in the general solution, thereby yielding a unique particular solution that fits the given initial condition.

Critical Point Analysis

This concept involves finding points at which the derivative of a function is zero or undefined, which are potential candidates for local extrema (maximum or minimum values). By analyzing the behavior of the function around these critical points, one can determine the conditions under which the function attains its maximum or minimum values.

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