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 = logax)
⇒
⇒
⇒
⇒
(Assuming log(c2) = K a constant)
∴ The solution to the given differential equation is log(y2 + x2) + 2tan-1 = K