Consider the initial value problem dy/dt + 3y = 7, y(0) =...

Consider the initial value problem dy/dt + 3y = 7, y(0) = 0. a. Find a nonzero solution to the associated homogeneous differential equation. y = b. Find a particular solution to the nonhomogeneous differential equation. y = c. Find the solution to the initial value problem. y = Note: You can earn partial credit on this problem.

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00:01 First one is the given differential equation is dy divided by dt plus 2y equal to 7.

00:08 So, now coming to part a here we have to find the non -zero solution of the homogeneous differential equation.

00:15 We can say that homogeneous differential equation is given by dy divided by dt plus 2y equal to 0.

00:23 It can be written as dy divided by dt equal to minus 2y.

00:28 Now, this can be written as 1 divided by y into dy which is equal to minus 2 dt.

00:36 Now, take integration on both sides so we get 1 divided by y dy which is equal to integral minus 2 dt which is equal to log of y which is equal to minus 2 into t plus c1.

00:52 So, log of y equal to minus 2 t minus 2 c1.

00:58 Now, taking anti -log on both sides so we get e power log of y which is equal to e power minus 2 t minus e power minus 2 c1.

01:09 Now, assume new constant c equal to e power minus 2 c1.

01:14 Therefore, e power log of y equal to e power minus 2 t of c and y equal to c e power minus 2 t.

01:28 Hence, the homogeneous differential equation is y h of t equal to c e power minus 2 t.

01:38 Now, coming to part b here we have to find the particular solution of the given differential equation.

01:44 On the right hand side of the differential equation we have g of t equal to 7 which is nothing but a constant.

01:52 Hence, let us assume that particular solution has the form of given by y p of t which is equal to a.

01:59 Consider this is equation 1...

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