# Order and solutions to differential equations

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##### Intros
###### Lessons
1. Differential Equations Overview
2. Notation and Order of a Differential Equation
3. Solution to a Differential Equation
4. Particular Solutions
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##### Examples
###### Lessons
1. Finding the Order of a Differential Equation
What is the order for the following differential equations?
1. $y''' = {t^2} + 7$
2. $y''( t ) + 6y'( t ) + 8y( t ) = ln( t )$
3. $5\frac{{dy}}{{dt}} = \csc \left( {4t} \right) + \frac{{{d^2}y}}{{d{t^2}}}$
4. ${x^3}\frac{{{d^3}y}}{{d{x^3}}} + {x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} = 1$
2. Verifying Solutions
Show that the following functions is a solution to the differential equation:
1. $y = \cos( 5t )$ is a solution to $y'' + 25y =$0
2. $y = C\cos \left( {5t} \right)$ is a solution to $y'' + 25y =$0 where $C$ is a constant
3. $y = C{e^x}$ is a solution to $3y + y' + y'' - 3y''' = 2C{e^x}$ where $C$ is a constant
4. $y = \frac{{{x^4}}}{4} + {x^2}$ is a solution to $y''' = 6x$
3. Finding a Particular Solution
You are given the general solution as well as the initial condition. Find the particular solution which suits the following initial conditions:
1. $6{x^3} + 9{y^2} = C$ where $y\left( 0 \right) = 3$
2. $C{e^{3t}} = {y^2}$ where $y\left( 1 \right) = 1$
4. Integrating to Find the General Solution
Find the general solution of the differential equation$\;\frac{{dy}}{{dx}} = x{e^{{x^2}}}$.