Solving First Order Differential Equations By Eliminating The Dependent Variable (Part 2, Solving for x)

 The method of solving for x is also called the 'method of eliminating the independent variable', which I examine in this post.

 Given a first order differential equation which can be readily solved for x, it can be written in the form: 

(1) x=  f(y, p)  

Then the total derivative of this equation with respect to y is: 

(2) dx/dy = 1/p = ¶ f/ ¶ y   +  ¶ f/ ¶ p (dp/ dy)  

Which is an equation involving only y and p. 

If we can solve this differential equation to obtain:

(3)  F(y, p, c) = 0  

Then equations (1) - (3) form the general parametric solution of the original equation. Once again, the procedure is to eliminate p and obtain the general solution.

Example: 

Solve: p ² + 2 y - 2x

Solve for x:  x =    ½ (2 y + p ²) 

Differentiate:  dx/ dy = 1/p = 1 + p (dp/dy)

Simplify to:

( p ²/ p - 1 ) dp + dy = 0

Or:

(p +  1  +  1/ p - 1) dp  + dy = 0

Solve by finding a general solution:

½ p ² + p +  ln (p - 1) + y = c   

Replace dx/ dy with 1/p in original eqn.

Separate variables to find y in terms of p.

 1/ p   - p =  p (dp/dy)

(1 - p) / p = p (dp/dy)

dy =    (p ² /1 -  p ) dp

dx =  (1/p)  dy

Subst. into original DE (dx/ dy = 1/p) to get:

dx =  (1/p)  (p ² dp/ 1 -  p )

->    

dx =  p dp/ (1 - p)

We now integrate:

To get:

x = c - p -  ln (1 - p)

Which is the general solution for x.

Suggested Problem:

Solve for x by finding a general solution:

 py ² -   x   =   0