5.8: Homework- Initial Value Problems — Read

From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

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 $f(x) = \sqrt{A+2 x+1} - 1$, $f(0) = 4$.
  
  $Illustration$
  
  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 was last modified on Tue, 19 Oct 2021 06:45:32 GMT and 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.

Contents

Text