Reduction of Equations

Reduction of Equations
There are some equations from second order can be reduced to a first order equation and there are two cases;
Case 1: The variable y is missing, substituting y^ =p and then y^ =dp/dx .
Example1: Solve x (d^2 y)/(dx^2 )+dy/dx=0
By substituting y^ =p and then y^ =dp/dx
x dp/dx+p=0
dp/dx+p/x=0
Which is a linear eq.
The solution is
dy/dx=p
Then y=c_1 lnx+c_2

Case 2: The variable x is missing, substituting y^ =p and then
(d^2 y)/(dx^2 )=dp/dx=dp/dy dy/dx=p dp/dy
Example2: Solve (d^2 y)/(dx^2 )+y dy/dx=0
The equation be
p dp/dy+yp=0
Which is a first order eq. in which variables can be separated
Reduction of Equations
There are some equations from second order can be reduced to a first order equation and there are two cases;
Case 1: The variable y is missing, substituting y^ =p and then y^ =dp/dx .
Example1: Solve x (d^2 y)/(dx^2 )+dy/dx=0
By substituting y^ =p and then y^ =dp/dx
x dp/dx+p=0
dp/dx+p/x=0
Which is a linear eq.
The solution is
dy/dx=p
Then y=c_1 lnx+c_2

Case 2: The variable x is missing, substituting y^ =p and then
(d^2 y)/(dx^2 )=dp/dx=dp/dy dy/dx=p dp/dy
Example2: Solve (d^2 y)/(dx^2 )+y dy/dx=0
The equation be
p dp/dy+yp=0
Which is a first order eq. in which variables can be separated