Q.Family $y=Ax+A^3$ of curves is represented by the differential equation of degree:
(A) 1
(B) 2
(C) 3
(D) 4

Eliminate the parameter. From $y=Ax+A^3$, $\frac{dy}{dx}=A$. Substituting $A=\frac{dy}{dx}$ back:

$y=x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^3.$ …

Find the differential equation

The family $y=Ax+A^3$ has one arbitrary constant $A$. Differentiate once:

So $A=\frac{dy}{dx}$. Substitute this back into $y=Ax+A^3$:

Read off the degree …

Method: Degree of the DE of a Family Given by a Parameter

To get the differential equation of a family like $y=Ax+A^3$, eliminate the parameter by differentiation and substitution, then read the degree from the highest power of the highest-order derivative.

Steps

Step 1: Differentiate to express the parameter.

From $y=Ax+A^3$, $\frac{dy}{dx}=A$, so $A=\frac{dy}{dx}$.

Step 2: Substitute the parameter back.

Step 3: Read order and degree. …

Common Mistakes

Mistake 1: Confusing degree with order.

Why it's wrong: after eliminating $A$, only $\frac{dy}{dx}$ appears (order $1$), but it is raised to the third power, so the degree is $3$. Correct approach: order is the highest derivative; degree is its highest power — here $1$ and $3$ respectively.

Mistake 2: Forgetting to substitute $A=\frac{dy}{dx}$ back. …