Question

The normal form (5) of an nth-order differential equation is equivalent to (4) whenever both forms have exactly the same solutions. Make up a first-order differential equation for which$F(x,y,y\text{'})=0$ is not equivalent to the normal form$dy/dx=f(x,y)$ .

Short answer

The first order DE $y\text{'}y-xy=0$.

Step-by-step solution

Definition of the differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

Family of solutions.

Consider the first order ODE

$y\text{'}y-xy=0$ ......(1)

Its normal form is,

$\frac{dy}{dx}=x$.....(2)

Note that,$y\left(x\right)=0$is a solution to (1) but not of (2), so this is an example of a first order DE that is not equivalent to its normal form.

Hence, the first order DE $y\text{'}y-xy=0$.