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romanna [79]
3 years ago
10

Find a solution to y'(t) = te^-t satisfying the condition y(1) = 1.

Mathematics
1 answer:
Alex_Xolod [135]3 years ago
4 0

Answer:

y=-e^{-t}(t+1)+1+\frac{2}{e}

Step-by-step explanation:

The given differential equation is

y'(t)=te^{-t}

It can be written as

\frac{dy}{dt}=te^{-t}

dy=te^{-t}dt

Integrate both sides.

\int dy=\int te^{-t}dt

Apply ILATE rule on right side. Here, t is first function and e^{-t} is the second function.

y=t\int e^{-t}-\int (\frac{d}{dt}t\int e^{-t})

y=-te^{-t}-\int (1\times (-e^{-t}))         \int e^{-x}=-e^{-x}+C

y=-te^{-t}+\int e^{-t}

y=-te^{-t}-e^{-t}+C             .... (1)

Initial condition is y(1) = 1. It means at t=1 the value of y is 1.

1=-(1)e^{-t}-e^{-(1)}+C

1=-e^{-1}-e^{-1}+C

1=-2e^{-1}+C

1=-\frac{2}{e}+C

Add \frac{2}{e} on both sides.

1+\frac{2}{e}=C

Substitute the value of C in equation (1).

y=-te^{-t}-e^{-t}+1+\frac{2}{e}

y=-e^{-t}(t+1)+1+\frac{2}{e}

Therefore, the solution of given initial value problem is y=-e^{-t}(t+1)+1+\frac{2}{e}.

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