Question

P=cos(y-px) general solution of the equation

Answer 1

Answer is (⇒y=0 is a solution)

step by step explain

(The given equation can be written as px−y=sin

−1

p

Differentiating w.r.t x, both sides

(x−

1−p

2

1

)

dx

dp

=0⇒p=c or x=

1−p

2

1

Putting p=c in the equation we have cx−y=sin

−1

c

Also x=

1−p

2

1

⇒p=

x

x

2

−1

∴y=

x

2

−1

−sin

−1

x

x

2

−1

⇒y=0 is a solution

Answer 2

The general solution of the equation is y = cos^-1(c)+cx

Given:

Differential equation P=cos(y-px).

To Find:

The General solution of the given differential equation.

Solution:

P=cos(y-px).

⇒ y - px = cos^-1 (P)

⇒ y =px  + cos^-1(P)

Now, it is in the form of y= xP + f(P)

An equation of the form  y= xP + f(P) is known as Clairaut’s equation. Where P = dy/dx. When the equation is in the form of Clairaut’s equation. The general equation can be found by replacing the P in the differential equation with c  where 'c' is an arbitrary constant.

∴ The general solution of the equation is y = cos^-1(c)+cx

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