Solve the following equations:

Given Differential equation is :

⇒ ……(1)

Homogeneous equation: A equation is said to be homogeneous if f(zx,zy) = zⁿf(x,y) (where n is the order of the homogeneous equation).

Let us assume:

⇒

⇒

⇒

⇒ f(zx,zy) = z⁰f(x,y)

So, given differential equation is a homogeneous differential equation.

We need a substitution to solve this type of linear equation, and the substitution is y = vx.

Let us substitute this in (1)

⇒

We know that:

⇒

⇒

⇒

⇒

⇒

Bringing like variables on one side we get,

⇒

⇒

⇒

⇒

We know that:

and Also,

Integrating on both sides, we get,

⇒

⇒

(∵ LogC is an arbitrary constant)

⇒

(∵)

Since y = vx,

we get

⇒

(∵ xloga = loga^x)

⇒

⇒

⇒

⇒

(Assuming log(c²) = K a constant)

∴ The solution to the given differential equation is log(y² + x²) + 2tan^-1 = K