33–42. Solving initial value problems Solve the following init...

Question

33–42. Solving initial value problems Solve the following initial value problems.

y'(t) = 1 + eᵗ, y(0) = 4

Accepted Answer

Welcome back, everyone. Solve the initial value problem. X T equals 3 multiplied by the power of TT and X 0 equals 5. For this problem, we're going to write the differential form of the derivative, so that would be D X divided by DT on the left hand side, and on the right hand side we have 3 E to the power of 2T. We can separate the variables by multiplying both sides by DT. So we get the X is equal to 3E to the power of 2T multiplied by DT. And now what we're going to do is simply integrate both sides, right? We have successfully separated the variables, and now we can just integrate. On the left hand side, the integral of D X is just X. On the right hand side we can factor out 3 and integrate E to the power of 2 T. The integral of E to the power of 2T is E to the power of 2T divided by the slope of the linear term in the exponent. So we're dividing by 2 according to the reverse chain rule. Plus a constant of integration C, right, because we have an indefinite integral. So we have shown that X is equal to 3 halves each the power of 2T plus C, and we know that X of 0 is 5. Which means that X is 5 when T is 0. In other words, we're going to set X equals 5 and T is 0. To identify the constant of integration C. We get 5 equals 3 halves multiplied by E to the power of 2 multiplied by 0 is 0. Plus C So we get 5 equals 3 halves plus C, and then C is equal to 5 minus 3 halves. 5 can be written as 10 divided by 2, and 10 minus 3 is 7, so we get 7 halves, that's our constant C. So our particular solution is X equals 3 halves eats the power of 2T plus 7 halves. And we can write it on a single fraction, that would be 3E to the power of 2T plus 7, all divided by 2. Well done, thank you for watching.

33–42. Solving initial value problems Solve the following initial value problems.y'(t) = 1 + eᵗ, y(0) = 4

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33–42. Solving initial value problems Solve the following initial value problems.
y'(t) = 1 + eᵗ, y(0) = 4