Initial Value Problems

Solve the initial valu...

Question

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dv/dt = (1/2)sec t tan t, v(0) = 1

Accepted Answer

Welcome back everyone. Solve the initial value problem. DV divided by DT equals 1/3 sequin squat and V at pi divided by 4 is equal to 1/2. So to solve this differential equation, we can use separation of variables. Let's multiply both sides by dt, which gives us dV equals 1/3. 2 squared cdc and now we can integrate both sides. On the left hand side, the integral of dv is simply v, and on the right hand side we can factor out one third because it's a constant. And we can integrate sequin squared c dt. This is an elementary integral. Its value is equal to tangent c. So we get 1/3 tangent c plus a constant of integration c, right, because we have an indefinite integral. And now to find C, we know that if T is equal to pi divided by 4, then V is going to be 1/2. So we're going to substitute V equals 1/2 on the left hand side. And on the right hand side t is going to be pi divided by 4, so we get 1/2 equals 1/3. Tangent of pi divided by 4 plus c. Which is 1/3 multiplied by The tangent of pi divided by 4 is 1, so we just have 1/3 plus c. So now to sulfur C, we're going to take 1/22 and subtract 1/3. Which is basically. We Divided by 6 minus 2 divided by 6, and this is equal to 1/6. Therefore, the final solution is going to be V of T. Is equal to 1/3. Tangent sea. + 1/6 That's our final solution, and thank you for watching.

Initial Value ProblemsSolve the initial value problems in Exercises 71–90.dv/dt = (1/2)sec t tan t, v(0) = 1

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Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dv/dt = (1/2)sec t tan t, v(0) = 1