# Find a particular solution to y'' +4y=6\sin(2t) .

## Question:

Find a particular solution to {eq}y'' +4y=6\sin(2t) {/eq}.

## Differential Equation in Calculus:

The most general linear differential equation of n-th order is {eq}\dfrac {d^{n}y}{dt^{n}}+P_{1}\dfrac {d^{n-1}y}{dt^{n-1}}+\ldots - - +P_{n}y=Q {/eq} can also be written as {eq}\left\{ D^{n}+P_{1}D^{n-1}+\ldots +P_{n}\right\} y=2 {/eq} where the operator {eq}D {/eq} is for {eq}\dfrac {d}{dt},D^2 {/eq} for {eq}\dfrac {d^{2}}{dt^{2}},... {/eq}

The particular solution of a differential equation can be obtained by particular values in the general solution. In some problems, the Particular solution does not satisfy the initial condition.

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{eq}y''+4y=6 \sin(2t) {/eq}

We can write the differential equation as

{eq}D^{2}+4D=6 \sin(2t) {/eq}

The particular solutiion {eq}y_{p}=\dfrac... Differential Calculus: Definition & Applications

from

Chapter 13 / Lesson 6
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This lesson is an introduction to differential calculus, the branch of mathematics that is concerned with rates of change. If you ever wanted to know how things change over time, then this is the place to start!