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MAT244––Home Assignments / Question Regarding Homework Assignment 2.1 #30
« on: January 14, 2018, 11:58:18 AM »
30) Find the value of y0 for which the solution of the initial value problem
$$
y′−y=1+3\sin(t)\qquad y(0)=y_0
$$
remains finite as $t\to \infty$.
It's clearly to see that the integrating factor is $e^{-t}$. OK. V.I.
Then $y(t) = e^{t}\int e^{-t } \bigl(1 + 3\sin(t)\bigr)\,dt + ce^t$, I fixed it
whichcan be easily solved as implies
$$
y(t) = - 1 - \frac{3}{2}\bigl(\sin(t) + \cos(t)\bigr) + ce^t
$$
But how can we interpret the term "remains finite as $t\to \infty$" into algebraic language for solving this initial value problem?
$$
y′−y=1+3\sin(t)\qquad y(0)=y_0
$$
remains finite as $t\to \infty$.
It's clearly to see that the integrating factor is $e^{-t}$. OK. V.I.
Then $y(t) = e^{t}\int e^{-t } \bigl(1 + 3\sin(t)\bigr)\,dt + ce^t$, I fixed it
which
$$
y(t) = - 1 - \frac{3}{2}\bigl(\sin(t) + \cos(t)\bigr) + ce^t
$$
But how can we interpret the term "remains finite as $t\to \infty$" into algebraic language for solving this initial value problem?