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

• jaredmt
In summary, in order to find the singular solution for the equation y = x(dx/dy) - (1/4)(dy/dx)^4, the first step is to let dy/dx = p. Then, by differentiating both sides and setting p' = 0 and x - p^3 = 0, you can find the singular solution. It is important to review your class notes to ensure that your results meet the definitions.
jaredmt

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

jaredmt said:

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.

1. How do you solve for the singular solution of y = x(dx/dy) - (1/4)(dy/dx)^4?

To solve for the singular solution of this equation, you can use the method of separation of variables. This involves isolating the variables dx and dy on one side of the equation and integrating both sides with respect to x and y, respectively. This will result in a general solution, which can then be simplified to find the singular solution.

2. Why is it important to find the singular solution of this equation?

The singular solution represents the special case where the derivative of y with respect to x, or dy/dx, is equal to zero. This solution is important because it can provide critical information about the behavior of the function, such as the location of critical points or inflection points.

3. Are there any other methods for solving this equation?

Yes, there are other methods such as using an integrating factor or using substitution to transform the equation into a separable form. However, the method of separation of variables is the most commonly used and straightforward approach for solving this type of equation.

4. Can this equation have multiple singular solutions?

No, this equation can only have one singular solution. This is because the singular solution represents the point where the derivative of y with respect to x is equal to zero, and there can only be one such point on a curve.

5. How can I check if my solution for the singular solution is correct?

You can check your solution by plugging it back into the original equation and seeing if it satisfies the equation. If it does, then your solution is correct. You can also graph the equation and the singular solution to visually confirm if they intersect at the point where dy/dx is equal to zero.

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