Question

Use the integrating factor method to find y solution of the inital value problem ty'=y-t^2sin(2t)

Answer 1

The integrating factor method is used to solve linear first-order differential equations. To apply this method to the given initial value problem, start by rewriting the equation in standard form. Then, identify the integrating factor and multiply both sides of the equation by it. Integrate both sides and solve for y to find the solution.

Step-by-step explanation:

The integrating factor method is used to solve linear first-order differential equations by multiplying both sides of the equation by an integrating factor. In this case, we have the initial value problem: ty' = y - t^2sin(2t). To find the solution using the integrating factor method, follow these steps:

1. First, rewrite the equation in standard form: ty' - y = -t^2 sin(2t).
2. Next, identify the integrating factor, which is e^(∫-dt/t) = 1/t.
3. Multiply the entire equation by the integrating factor: (1/t)(ty') - (1/t)y = -(1/t)t^2sin(2t).
4. Simplify the left side to get d(ty)/dt - y/t = -(1/t)t^2sin(2t).
5. Integrate both sides with respect to t: ∫d(ty)/dt dt - ∫y/t dt = -∫(1/t)t^2sin(2t) dt.
6. Simplify to get ty - y + C = -∫t sin(2t) dt, where C is the constant of integration.
7. Finally, solve for y by dividing both sides by t and rearranging the equation: y = (1 - ty)/t + C/t + ∫(sin(2t)/t) dt.So the solution to the initial value problem is y = (1 - ty)/t + C/t + ∫(sin(2t)/t) dt, where C is a constant.