Exact differential equations

Intros
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Lessons
  1. Partial Differentiation Review
  2. What are Exact Differential Equations and how do we solve them?
Examples
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Lessons
  1. Partial Differentiation
    Find the first order partial derivative of every variable for the following functions:
    1. f(x,y)=3x2−3y+2 f(x,y)=3x^2-3y+2
    2. N(y,z)=2y2z N(y,z)=2y^2 z
  2. Determining what is an Exact Differential Equation
    Solve the following differential equation

    2xy2−y2+(2x2y−2xy)dydx=02xy^2-y^2+(2x^2 y-2xy) \frac{dy}{dx}=0

    Using the equation: Ψ(x,y)=x2y2−xy2=2\Psi(x,y)=x^2 y^2-xy^2=2
  3. Solving Exact Equations
    Solve the following differential equation

    3exy−12y2+(3ex−xy)dydx=03e^x y- \frac{1}{2} y^2+(3e^x-xy) \frac{dy}{dx}=0

    With initial conditions y(0)=2y(0)=2
    1. Solve the following differential equation

      3x2y+2y2=3x+1+y′(2−3y−4xy−x3)3x^2 y+2y^2=3x+1+y' (2-3y-4xy-x^3)

      With initial conditions y=1,x=1y=1, x=1
    2. Verify that the solution you found is in fact the solution to the above differential equation (may be useful to do on a test).