1 INITIAL-VALUE PROBLEM
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1 INITIAL-VALUE PROBLEM

[From: ] [author: ] [Date: 11-08-10] [Hit: ]
-The equation is first order linear.exp(∫ -2t dt) = e^(-t²).d/dt [e^(-t²) v] = 3t².e^(-t²) v = t^3 + C ==> v(t) = t^3 e^(t²) + C e^(t²).Apply the initial condition to get 5 = 0 + C ==> C = 5.v(t) = (t^3+ 5)e^(t²).......
Solve the initial-value problem:

dv/dt -2tv = (3t²)(e^t²) , v(0) = 5


Thanks in advance!

-
The equation is first order linear. Use the integrating factor

exp(∫ -2t dt) = e^(-t²).

Multiply through and the left side is a product rule:

d/dt [e^(-t²) v] = 3t².

Integrate

e^(-t²) v = t^3 + C ==> v(t) = t^3 e^(t²) + C e^(t²).

Apply the initial condition to get 5 = 0 + C ==> C = 5. So

v(t) = (t^3+ 5)e^(t²).
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