Which one of the following is a general solution of the differential equationy =2px + p2y ?a)2yc - x2 +c2 = 0b)2xc +y2 - c2=0c)2xc - y2+ c2 = 0d)yc + x2+c2 = 0Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Some comments before the solution.
1. If the given equation in x, y and p can be solved for y, then we shall have
y = f(x,p) ...(i)
Differentiate (i) w.r. t. x. This will lead to a differentia! equation in p and x. Solve it for p and then eliminate p from the resulting equation and (i).
2. If the given equation in x, y and p can be solved for x, then we shail have
x = f(y, p) .......(ii)
Differentiate (ii), w.r. t. y and follow the same procedure as in case - i.
Now solution of the problem 82.
The given differential equation is
y = 2 px + p²y ...... (iii)

... (iv)
which is solved for x, i.e. it expresses x as a function of y and p.
Now differentiating (iv) w.r. t. y, we get

Now eliminating p from equations (iii) and (v), we get

or y² = 2cx + c²

Answer

To solve the given differential equation, we need to find a general solution that satisfies the equation.

The given differential equation is: y = 2px - p^2y

To solve this equation, we can rearrange it to isolate the terms involving y on one side:

y + p^2y = 2px

Factor out y on the left side:

y(1 + p^2) = 2px

Divide both sides by (1 + p^2):

y = (2px) / (1 + p^2)

Now, let's simplify this expression further.

We can rewrite 2px as 2xc since p represents the derivative of x with respect to y.

Therefore, the general solution of the given differential equation is:

y = (2xc) / (1 + p^2)

Now, let's rewrite this expression in terms of x and y.

p = dx/dy, so we can rewrite it as:

p = (dx/dy) = (dx)/(dy/dx) = (dx/dy) / (1/(dx/dy))

Therefore, p^2 = ((dx/dy) / (1/(dx/dy)))^2 = ((dx/dy)^2) / (1/(dx/dy))^2 = (dx^2)/(dy^2)

Substituting this back into the equation, we get:

y = (2xc) / (1 + (dx^2)/(dy^2))

Now, let's simplify this expression further.

Multiply the numerator and denominator by (dy^2):

y = (2xc * dy^2) / (dy^2 + dx^2)

Now, we can rewrite dy^2 + dx^2 as d(x^2 + y^2):

y = (2xc * dy^2) / d(x^2 + y^2)

Cancel out the d terms:

y = (2xc * y^2) / (x^2 + y^2)

Multiply both sides by (x^2 + y^2):

y(x^2 + y^2) = 2xc * y^2

Rearrange the terms:

y(x^2 + y^2 - 2xc * y) = 0

This is the general solution of the given differential equation.

Therefore, the correct option is C) 2xc - y^2 + c^2 = 0.

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