Find the general solutions of the following equation Y = px + a/p.

1.	Find the general solutions of the following equation Y = px + a/p.
Answer» The given equation is y = px + (a/p), ......(1) Which is in Clairaut's form. So replacing p by c in (1) the solution is y = cx + (a/c) or c²x - yc + a = 0. ........(2) Now, c - discriminant relation of (2) is B² - 4AC = 0; i.e., (-y)² - 4xa = 0 or y² = 4ax .......(3) Now, y² = 4ax gives 2y(dy/dx) = 4a or p = 2a/y. Putting this value of p in (1), we get y = (2ax)/y + (y/2) or y² = 4ax which is true by (3) satisfies (1) so y² = 4ax is the required singular solution.

Answer»

The given equation is y = px + (a/p), ......(1)

Which is in Clairaut's form. So replacing p by c in (1) the solution is

y = cx + (a/c) or c²x - yc + a = 0. ........(2)

Now, c - discriminant relation of (2) is

B² - 4AC = 0; i.e.,

(-y)² - 4xa = 0 or y² = 4ax .......(3)

Now, y² = 4ax gives 2y(dy/dx) = 4a or p = 2a/y. Putting this value of p in (1), we get y = (2ax)/y + (y/2) or y² = 4ax which is true by (3) satisfies (1) so y² = 4ax is the required singular solution.

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Find the general solutions of the following equation Y = px + a/p.

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