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# 5.8: Homework- Initial Value Problems

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1. Verify that the given solution to each differential equation is correct, and solve for the free parameter.
1. Differential equation $$f'(t) = f(t) + 3$$, solution $$f(t) = A e^t - 3$$, $$f(0) = 4$$.

\begin{align*} f'(t) & = f(t) + 3 \\ \frac{d}{dt} (A e^t - 3) & = (A e^t - 3) + 3 \\ A e^t & = A e^t. \end{align*}

If $$f(0) = 4$$, then $$A = 7$$.

ans
2. Differential equation $$f'(t) = 2 f(t) - 2$$, solution $$f(t) = A e^{2t} + 1$$, $$f(0) = 0$$.

\begin{align*} f'(t) & = 2 f(t) - 2 \\ \frac{d}{dt} (A e^{2t} + 1) & = 2(A e^{2t} + 1) - 2 \\ 2 A e^{2t} & = 2 A e^{2t} + 2 - 2 \\ 2 A e^{2t} & = 2 A e^{2t} \end{align*}

If $$f(0) = 0$$, then $$A = -1$$.

ans
3. Differential equation $$f'(x) = \frac{1}{f(x) + 1}$$, solution $$f(x) = \sqrt{A+2 x+1} - 1$$, $$f(0) = 4$$.

If $$f(0) = 4$$, then $$A = 24$$.

ans
4. Differential equation $$f'(t) = (f(t))^2 + f(t)$$, solution $$f(t) = -\frac{Ae^{t}}{Ae^t - 1}$$, $$f(0) = 3$$.

\begin{align*} f'(t) & = (f(t))^2 + f(t) \\ \frac{d}{dt} \left( -\frac{Ae^{t}}{Ae^t - 1} \right) & = \left(-\frac{Ae^{t}}{Ae^t - 1}\right)^2 + \left( -\frac{Ae^{t}}{Ae^t - 1} \right) \\ -\frac{(A e^t - 1)Ae^t - Ae^t(Ae^t)}{(A e^t - 1)^2} & = \frac{(A e^t)^2}{(A e^t -1 )^2} - \frac{Ae^t}{A e^t - 1} \\ -\frac{(A e^t)^2 - Ae^t - (Ae^t)^2}{(A e^t - 1)^2} & = \frac{(A e^t)^2}{(A e^t -1 )^2} - \frac{Ae^t}{A e^t - 1} \cdot \frac{(A e^t - 1)}{(A e^t - 1)} \\ -\frac{- Ae^t}{(A e^t - 1)^2} & = \frac{(A e^t)^2}{(A e^t -1 )^2} - \frac{(A e^t)^2 - Ae^t}{(A e^t - 1)^2} \\ \frac{Ae^t}{(A e^t - 1)^2} & = \frac{(A e^t)^2}{(A e^t -1 )^2} + \frac{-(A e^t)^2 + Ae^t}{(A e^t - 1)^2} \\ \frac{Ae^t}{(A e^t - 1)^2} & = \frac{(A e^t)^2 - (Ae^t)^2 + Ae^t}{(A e^t -1 )^2} \\ \frac{Ae^t}{(A e^t - 1)^2} & = \frac{Ae^t}{(A e^t -1 )^2} \end{align*}

If $$f(0) = 3$$, then $$A = 1.5$$.

ans

This page titled 5.8: Homework- Initial Value Problems is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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