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 on the same side we get,
⇒
⇒
⇒
We know that
Integrating on both sides we get,
⇒
⇒
(∵ logC is an arbitrary constant)
⇒ log(1-2v2) = -4logx + 4logC
⇒ log(1–2v2) = -logx4 + logC4
(∵ xloga = logax)
⇒
(∵ )
Applying exponential on both sides we get,
⇒
Since y = vx, we get,
⇒
⇒
⇒
⇒
Cross multiplying on both sides we get,
⇒ x2(x2–2y2) = c4
⇒ x4–2x2y2 = c4
∴ The solution for the given differential equation is x4–2x2y2 = C4.