# Find the function y(t) that satisfies the differential equation dy/dt - 2ty = -12t^2 e^(t^2) and...

## Question:

Find the function {eq}y(t) {/eq} that satisfies the differential equation {eq}\; \frac{\mathrm{d}y}{\mathrm{d}t} - 2ty = -12t^2 e^{t^2} \; {/eq} and the condition {eq}\; y(0) = 1 {/eq}.

## Solution of the Differential Equation:

The given differential equation is first order linear differential equation. To find the solution of the differential equation first we find the integrating factor of the differential equation as follows.

{eq}\text{I.F}=e^{\int P(t)dt}=e^{\int (-2t) dt}=e^{-t^2} {/eq}

Then the solution of the differential equation will be written in the form of

{eq}y\times \textbf{I.F}=\int Q(t)\times \textbf{I.F}dt+c {/eq}

Now, we find the initial condition to evaluate the constant {eq}c {/eq}

## Answer and Explanation:

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Consider the differential equation

{eq}\frac{\mathrm{d}y}{\mathrm{d}t} - 2ty = -12t^2 e^{t^2},\quad y(0) = 1 {/eq}

Compare the differential...

See full answer below.

First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.