Differential Equations

Differential Equations

1. La Place

   • By Definition
   • Transforms
   • Derivative Transforms
   • 1st Translation Theorem
   • 2nd Translation Theorem
   • Unit Step
   • Transform Derivatives
   • Transform Integrals (Convolution)

2. First Order DE

   • Classification
     • By Type
       • Ordinary
       • Partial
     • By Order
       • Highest order Derivative
     • By Linearity
       • Linear
       • Non Linear
   • Methods of Solution
     • Separable Variables
       • (P) dx = (Q) dy
         • Integrate both sides and find Y
     • Linear Equation
       • dy/dx+P(x)Y=Q(x)
         • Y= Integral(e^(IntegralPdx) Q dx)/e^(IntegralPdx)
     • Exact Equation
       • If dP/dy=dQ/dx
         • C= Integral(P(x,y)Dx) + Integral(Q(x,y)Dy)
     • Integrating Factor
       • Not Exact, find U and multiplicate
         • U(x)= Integral((Pdy-Qdx)/Qdx))
         • U(y)= Integral((Qdx-Pdy)/Pdy))
     • Bernoulli
       • Find U = y^-n
         • Substitute variables to find Linear Standard Form
           • U= Integral(e^(IntegralPdx) Q dx)/e^(IntegralPdx)
             • Substitute U and Y, find Y

3. High Order DE

   • Superposition or Linear Combination
   • Homogenous = 0
   • Non Homogenous = f(x)
   • Methods of Solution
     • Reduction of Order
     • Constant Coefficients
     • Undetermined Coefficients
     • Variation of Parameters
     • Cauchy - Euler