Answer to Question #148145 in Differential Equations for Nikhil Singh

Question

The differential equation
(y.lnx-1)ydx=xdy is a linear differential equation of first order and first degree.
True or false with full explanation

Answer 1

FALSE

Though the D.E above is of first and first degree, it is not a linear DE.

EXPLANATION

The equation above can be written as $x\frac{dy}{dx}+y=y^2\ln x.$

The highest derivative there is $\frac{dy}{dx}$ and this is first order.

Also, the degree of the highest derivative is 1. This make it first degree.

A DE is said to be linear if there is no product of the dependent variable(function) and it's derivatives and neither the function nor its derivatives occurs to any power other than first power. But from the re-written equation, we see that $y$ has the power of 2. Hence, the DE is Non-linear.

With this, the expression is false.