Question

Family $y = Ax + {A^3}$ of curve represented by the differential equation of degree:
(A) three
(B) two
(C) one
(D) none of these

Answer 1

Hint : To find the degree of the family of curves we first find the corresponding differential equation associated with it. Differential equations can be found by eliminating the constant (A) present in the given family of curves by calculating the derivative of the given curve $y = Ax + {A^3}$ .

Complete step-by-step answer :
To find the degree of the differential equation we first frame a differential equation and then using it we find the degree of the differential equation of the given family of curves.
In a given curve we have one constant ‘A’ to find a differential equation for this curve, we eliminate ‘A’ from it by differentiating the given curve w.r.t. ‘x’.
Differentiating $y = Ax + {A^3}$ w.r.t. ‘x’ we have
$\dfrac{{dy}}{{dx}} = A(1) + 0$ $\because \dfrac{d}{{dx}}(x) = 1,\,\,\,\dfrac{d}{{dx}}\left( A \right) = 0(A\,\,being\,\,a\,\,\cos n\tan t)$
Or
$\dfrac{{dy}}{{dx}} = A$ , using this value of A in given family of curve we have
$y = \left( {\dfrac{{dy}}{{dx}}} \right)x + {\left( {\dfrac{{dy}}{{dx}}} \right)^3}$
From above we can say that $y = \left( {\dfrac{{dy}}{{dx}}} \right)x + {\left( {\dfrac{{dy}}{{dx}}} \right)^3}$ is the required differential equation of given family of curve $y = Ax + {A^3}$ .
Now, to find the degree of the family of curves we find the degree of the differential equation formed in the above step.
We know that the degree of the differential equation is the highest power of the highest order of derivatives.
Hence, from above we see that the highest order of derivative in the differential equation is $\dfrac{{dy}}{{dx}}$ and its highest power is three.
Therefore, we can say that degree of given family of curve $y = Ax + {A^3}$ associated with differential equation $y = \left( {\dfrac{{dy}}{{dx}}} \right)x + {\left( {\dfrac{{dy}}{{dx}}} \right)^3}$ is $3$

Note : Students must be careful while calculating the degree of differential equation that degree is always taken from power of highest order derivative not as highest power of involved in differential equation. For example
Degree of differential equation $\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) + {\left( {\dfrac{{dy}}{{dx}}} \right)^2} + y = 0$ is $1$ but not $2$ as power of second order derivative or highest derivative in given differential equation.