Solving for Singular Solution of y = x(dx/dy) - (1/4)(dy/dx)^4

[jaredmt]

Homework Statement

find singular solution:
y = x(dx/dy) - (1/4)(dy/dx)^4

Homework Equations

The Attempt at a Solution

ok i let dy/dx = p and ended up with:
y = xc - (1/4)c^4

and my professor says i got that much correct but apparently i didnt find the formula for singular solution. what am i supposed to do now? i thought that was the formula but i guess not

 

[Unco]

Homework Statement

find singular solution:
y = x(dx/dy) - (1/4)(dy/dx)^4

The Attempt at a Solution

ok i let dy/dx = p and ended up with:
y = xc - (1/4)c^4

and my professor says i got that much correct but apparently i didnt find the formula for singular solution. what am i supposed to do now? i thought that was the formula but i guess not

Let's assume you meant to write: y = x(dy/dx) - (1/4)(dy/dx)^4. Writing y' = dy/dx, that is y = xy' - (1/4)(y')^4. (*)

Then differentiating both sides with respect to x gives y' = y' + xy'' - y''(y')^3, so if y' = p, then p = p + xp' - p'p^3, i.e., 0 = p'(x - p^3).

The p'=0 case leads to the "general solution" (cf. plugging y'=p=c into (*) gives the result you have written); the x - p^3 = 0 case leads to the "singular solution". Of course, you should check your results do meet the definitions.

Review your class notes!