Order and solutions to differential equations

0/3
?
Intros
Lessons
  1. Differential Equations Overview
  2. Notation and Order of a Differential Equation
  3. Solution to a Differential Equation
  4. Particular Solutions
0/11
?
Examples
Lessons
  1. Finding the Order of a Differential Equation
    What is the order for the following differential equations?
    1. y=t2+7y''' = {t^2} + 7
    2. y(t)+6y(t)+8y(t)=ln(t)y''( t ) + 6y'( t ) + 8y( t ) = ln( t )
    3. 5dydt=csc(4t)+d2ydt25\frac{{dy}}{{dt}} = \csc \left( {4t} \right) + \frac{{{d^2}y}}{{d{t^2}}}
    4. x3d3ydx3+x2d2ydx2+xdydx=1{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)y = \cos( 5t ) is a solution to y+25y=y'' + 25y = 0
    2. y=Ccos(5t)y = C\cos \left( {5t} \right) is a solution to y+25y=y'' + 25y = 0 where CC is a constant
    3. y=Cexy = C{e^x} is a solution to 3y+y+y3y=2Cex3y + y' + y'' - 3y''' = 2C{e^x} where CC is a constant
    4. y=x44+x2y = \frac{{{x^4}}}{4} + {x^2} is a solution to y=6xy''' = 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. 6x3+9y2=C6{x^3} + 9{y^2} = C where y(0)=3y\left( 0 \right) = 3
    2. Ce3t=y2C{e^{3t}} = {y^2} where y(1)=1y\left( 1 \right) = 1
  4. Integrating to Find the General Solution
    Find the general solution of the differential equation  dydx=xex2\;\frac{{dy}}{{dx}} = x{e^{{x^2}}}.