Many times in order to solve a differential equation, all that's needed is to eliminate the dependent variable (y or x). To that end, we look first at solving DEs for y, then for x.
If a first order DE can be solved for y, they n it can be written in the form: y = f(x, p)
The total derivative of this equation with respect to x is:
dy/dx = p = ¶ f/ ¶ x + ¶ f/ ¶ p (dp/ dx)
An expression involving only x and p. The objective now is to solve this equation to obtain:
F(x, p, c) = 0
To obtain the general solution of the original differential equation we need to form a pair of parametric equations with p as the parameter. Then p can be eliminated to obtain a solution in x and y along with an arbitrary constant.
Example: Solve:
p 2 + 2 xp - 2y = 0
(1) Solve for y to obtain:
y = ½ p 2 + 2 xp
2) Differentiate with respect to x:
dy/dx = p = p (dp/ dx) + x (dp/ dx) + p
Or: (p + x) (dp/ dx) = 0
From this result we see:
Either: dp/ dx = 0 or: p + x = 0
Hence: p = c or p = -x
3) Eliminate p between the last equations and the solution for y:
y = ½ c 2 + c x
y = ½ x 2 - x 2 = - ½ x 2
Both of which satisfy the original differential equation. However, note that the 2nd solution does not contain c (an arbitrary constant) so is not the general solution. Rather it is what we call a singular solution.
Suggested Problem:
Solve by finding a general solution:
px 2 - y = 0
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