Review 6: Now You Try III

Exercise 1: Find the particular solution to the initial value problem

$LaTeX: \begin{equation*} \begin{cases} \frac{dy}{dx} = 7y^2x^3\\ y(0) = 5. \end{cases} \end{equation*}$ $\begin{equation*} \begin{cases} \frac{dy}{dx} = 7y^2x^3\\ y(0) = 5. \end{cases} \end{equation*}$

Note: Assume that the solution we saw in Now You Try I, Exercise I, is the general solution: $LaTeX: y = \dfrac{-1}{\frac{7}{4}x^4+C}.$ $y = \dfrac{-1}{\frac{7}{4}x^4+C}.$

Solution:

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We need to find the value of LaTeX: C

$C$ that ensures that the initial condition LaTeX: y(0) = 5

$y(0) = 5$ is satisfied. Plugging in LaTeX: x= 0

$x= 0$ into the general solution, we have $LaTeX: y(0) = \frac{-1}{C} = 5.$

$y(0) = \frac{-1}{C} = 5.$ Thus, we need LaTeX: C = -1/5.

$C = -1/5.$

We conclude that the particular solution to the initial value problem is $LaTeX: y(x) = \dfrac{-1}{\frac{7}{4}x^4-\frac{1}{5}}.$

$y(x) = \dfrac{-1}{\frac{7}{4}x^4-\frac{1}{5}}.$