First-Order Linear Equations
Solve the differential equa...

Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Accepted Answer

Welcome back everyone. Solve the differential equation sin x dyy divided by dx plus y cosin x equals sin x, and x is between 0 and pi not inclusive. So for this problem, generally we use the integrating factor, but let's notice that the equation is already given in its correct form for us to get the solution. So what do we mean by that? Essentially notice that the equation can be written as sinx multiplied by y derivative plus y multiplied by sin x derivative because sin x derivative is cosine x and this is equal to sin x. On the left hand side we can use the product rule of differentiation since we have y multiplied by sin x derivative, right. According to the product rule, this is going to be Y X plus Y X, right? And this is equal to sin x. Now we can rewrite it in the differential form, so that would be d divided by dx off y multiplied by sin x is equal to sin x, and we can multiply both sides by dx to get d off y multiplied by sin x is equal to sin x dx, which basically allows us to integrate both sides. On the left hand side we get y multiplied by sin x. That's the integral of d of y sin x. We're basically taking the integral of the differential to get y sin x. And on the right hand side we want to integrate sin x, which is negative cosine x, and we're going to add a constant of integration c. So now let's go ahead and solve for y. We're going to divide both sides by sin x. We get negative cosine x divided by sin x, which is negative cotangent x, plus c divided by sin x. And 1 divided by sin x is cosequin, so we can write c multiplied by cosequent x. So the final solution is y equals negative cotangent x plus c multiplied by cosequent x. Thank you for watching.

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

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First-Order Linear Equations
Solve the differential equations in Exercises 1–14.

tan θ dr/dθ + r = sin²θ, 0 < θ < π/2