Q. 33.7( 3 Votes )
Solve the following equations:
Given differential equation can be written as:
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:
Integrating on both sides, we get,
⇒ log(v2 + 1) = -logx + logC (∵ LogC is an arbitrary constant)
Since y = vx,
Applying exponential on both sides, we get,
Cross multiplying on both sides we get,
⇒ y2 + x2 = Cx
∴ The solution for the given differential equation is y2 + x2 = Cx.
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