Order and solutions to differential equations

Order and solutions to differential equations

In this lesson, we will look at the notation and highest order of differential equations. To find the highest order, all we look for is the function with the most derivatives. After, we will verify if the given solutions is an actual solution to the differential equations. We do this by simply using the solution to check if the left hand side of the equation is equal to the right hand side. Lastly, we will look at an advanced question which involves finding the solution of the differential equation.

Lessons

We say that:
$y' (x) = \frac{dy}{dx}$ or $y'(t) = \frac{dy}{dt}$

Where:
1. $y'(x)$ is the first derivative of the function y in terms of $x$.
2. $y'(t)$ is the first derivative of the function y in terms of $t$.
• 1.
Differential Equations Overview
a)
Notation and Order of a Differential Equation

b)
Solution to a Differential Equation

c)
Particular Solutions

• 2.
Finding the Order of a Differential Equation
What is the order for the following differential equations?
a)
$y''' = {t^2} + 7$

b)
$y''( t ) + 6y'( t ) + 8y( t ) = ln( t )$

c)
$5\frac{{dy}}{{dt}} = \csc \left( {4t} \right) + \frac{{{d^2}y}}{{d{t^2}}}$

d)
${x^3}\frac{{{d^3}y}}{{d{x^3}}} + {x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} = 1$

• 3.
Verifying Solutions
Show that the following functions is a solution to the differential equation:
a)
$y = \cos( 5t )$ is a solution to $y'' + 25y =$0

b)
$y = C\cos \left( {5t} \right)$ is a solution to $y'' + 25y =$0 where $C$ is a constant

c)
$y = C{e^x}$ is a solution to $3y + y' + y'' - 3y''' = 2C{e^x}$ where $C$ is a constant

d)
$y = \frac{{{x^4}}}{4} + {x^2}$ is a solution to $y''' = 6x$

• 4.
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:
a)
$6{x^3} + 9{y^2} = C$ where $y\left( 0 \right) = 3$

b)
$C{e^{3t}} = {y^2}$ where $y\left( 1 \right) = 1$

• 5.
Integrating to Find the General Solution
Find the general solution of the differential equation$\;\frac{{dy}}{{dx}} = x{e^{{x^2}}}$.