# Find the solution to the differential equation, subject to the given initial condition. P*dP/dt =...

## Question:

Find the solution to the differential equation, subject to the given initial condition.

{eq}P\frac{\mathrm{d} P}{\mathrm{d} t} = 1, \; P(0) = 3 {/eq}

## Initial Value Problem:

An initial value problem is a differential equation given with a specific initial condition. We take the following steps to solve this problem

1 ) Solve the differential equation to get a general solution

2) USe the differential equation to get a particular solution or to find the integration constant.

## Answer and Explanation: 1

{eq}P\frac{\mathrm{d} P}{\mathrm{d} t} = 1\\ Pd\mathrm{P} = \mathrm{d}t\\ \int Pd\mathrm{P} = \int \mathrm{d}t\\ \frac{P^2}{2} = t+C {/eq}

We are given

{eq}P(0) = 3\\ \frac{3^2}{2} = 0 + C\\ C = \frac{9}{2}\\ \implies P = \pm 2 \Big(\sqrt{t+ \frac{9}{2}}\Big) {/eq}

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Chapter 11 / Lesson 13Learn to define the initial value problem and initial value formula. Learn how to solve initial value problems in calculus. See examples of initial value problems.

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