First Order Differential Equation

Differential Equation

1. Ordinary differential Equation(ODE)
2. Partial Differential Equation(PDE)

Property

1. Order of DE
2. Degree of DE

Condition for DE

1. Initial value 
2. Boundary value

Type of Differential Equation

1. Linear and non-linear
2. Homogeneous and non-homogeneous
3. First order differential equation
4. Second order differential equation
5. n-th order differential equation.
6. Bernoulli equation

First order differential equation

1. Standard form of FODE   y'=f(x,y)
2. Differential form M(x,y)dx+N(x,y)dy=0
3. linear form y'+p(x)y=q(x)
4. Bernoulli y'+p(x)y=q(x)yⁿn
5. Homogeneous f(x,y)=f(tx,ty)
6. exact differential equation M_y(x,y)=N_x(x,y)
1. 

Solution 4 first ODE

1. Separable variables f(x)dx+g(y)dy=0
2. Reduction of Homogeneous 
1. substitute y=vx => y' = v+xv'   OR x=uy => x'=u+xu'
3. Exact differential equation
1. do integration g_x(x)=M(x,y) with respect to x => g(x,y)= I(M(x,y)dx+h(y)
2.  When do integration constant would depends on y function h(y)
3. this result need to do partial differential with respect to y and put equal to g_y(x,y)=N(x,y)
4. Integrate this and get solution for h(y) and put in step 1 equation.
4. Linear Differential equation. y'+p(x)y=q(x)
1. find integration factor
1. IF(x,y)=ePower(I(px)dx)
2. mulitply this integration factore with equation get this
1. yIF=I(IF*q(x))dx + C
5. Reduction of Bernoulli Equations
1. substitute z=y^1-n
2. Now we have linear equation.