Chapter 2

Chapter 2

1. Separation Variables

   • M(x,y)dx + N(x,y)dy =0
   • Steps to solve
     • Step 1: divide y and dy, x and dx into different side
     • Step 2: integrate both side and put the constant C
     • Step 3: simplify the equation

2. Homogeneous

   • dy/dx = x(dv/dx) +v
   • Steps to solve
     • Step 1: let y=xv and differentiate then substitute into equation
     • Step 2: solve the new equation using separation variable
     • Step 3: plug back the x or y then solve it

3. Exact Equation

   • M(x,y)dx + N(x,y)dy =0
   • Steps to solve
     • Step 1: test of exactness. differentiate M(x,y) with respect to y and N(x,y) with respect to x. if same answer its exact!
     • Step 2: integrate M(x,y) with respect to x OR N(x,y) with respect to y
     • Step 3: differentiate the equation that we have integrate it with different respect to.
     • Step 4: compare to the original equation to find h'(y) then integrate to find answer and substitute back to original equation

4. Integrating Factor

   • dy/dx + P(x)y = Q(x)
   • Steps to solve
     • Step 1: make the original equation in standard form
     • Step 2: integrating factor v(x) = e^(∫P(x)dx)
     • Step 3: multiply both side with v(x)
     • Step 4: solve the resultant exact equation

5. Substitution

   • M(x,y)dx + N(x,y)dy =0
   • Steps to solve
     • Step 1: take a term of the equation and replace with v
     • Step 2: solve dy/dx in terms of v and dv/dx
     • Step 3; substitute dv/dx and solve for v using I.F. method
     • Step 4: substitute back v into original term replaced and solve for y

6. Bernoulli's Equation

   • dy/dx+P(x)y=f(x) y^n
   • Steps to solve
     • Step 1: substitute v= y^(1-n)
     • Step 2: solve resulting ist order DE
     • Step 3: after solving for v determine the solution to original equation

7. Coefficients Linear

   • (a1 x+b1 y+c1 )dx + (a2 x+b2 y+c2 )dy=0
   • Steps to solve it
     • Two lines intersect
       • Step 1: let x=u+h and y=v+k
       • Step 2: transformed original equation to (a1 u+b1 v)du + (a2 u+b2 v)dv =0
       • Step 3: solve using homogeneous in u and v
     • Two lines are parallel
       • Step 1: put constant k in the equation (a1 x+b1 y+c1 )dx + [k〖(a〗2 x+b2 y)+c2]dy=0
       • Step 2: let v=a1x + b1y and differentiate with respect to y
       • Step 3: solve using separation variable