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rdioface
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Homework Statement
For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution.
Homework Equations
[itex] y' - 4ty = 1 [/itex]
[itex] y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds[/itex]
The Attempt at a Solution
I attempted to take [itex]\frac{d}{dt}[/itex] of y(t) as usual but
1. if I do not try bringing the [itex]\frac{d}{dt}[/itex] inside the integral I can do nothing because there is no elementary antiderivative of y(t).
2. if I do bring the [itex]\frac{d}{dt}[/itex] inside the integral, I can use the chain rule to get
[itex] y(t) = (-2) \int_{0}^{t} (-2t) e^{-2(s^{2}-t^{2})} ds[/itex]
but since my variable of integration is ds not dt, this doesn't allow me to use a u-substitution as I had hoped nor can I think of a way to relate ds and dt.
More or less I do not know how to take [itex]\frac{d}{dt}[/itex] of y(t) and I do not know any other ways to solve the problem.
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