Question

42. The differential equation for the parabola ya = 4ax, where a is a parameter, is :
(B) 2x
hr
2x dy
dr
(A) 2
= y
(C) 3
en
+ y = 0
dy
(D) y + x = 0​

Answer 1

Answer :

y = 2x•dy/dx

Note :

• An equation involving the derivatives of dependent variable with respect to independent variable and possibly x and y is called a differential equation .

• We can find the differential equation of a given curve by differentiating it and eliminating the arbitrary constants / parameters .

Solution :

Here ,

The given equation of parabola is ;

y² = 4ax ------(1)

Now ,

Differentiating eq-(1) both sides with respect to x , we get ;

=> dy²/dx = d(4ax)/dx

=> (dy²/dy)•(dy/dx) = 4a•dx/dx

=> 2y•dy/dx = 4a

=> a = (2y/4)•dy/dx

=> a = (y/2)•dy/dx

Now ,

Putting a = (y/2)•dy/dx in eq-(1) , we get ;

=> y² = 4•(y/2)•(dy/dx)•x

=> y = 2x•dy/dx

Hence ,

Required differential equation is :

y = 2x•dy/dx

Alternative method :

Here ,

The given equation of parabola is ;

=> y² = 4ax

=> y²/x = 4a

Now ,

Differentiating both sides with respect to x , we get ;

=> d(y²/x)/dx = d(4a)/dx

=> d[y²•(1/x)]/dx = 0

=> (1/x)•dy²/dx + y²•d(1/x)/dx = 0

=> (1/x)•2y•dy/dx - y²/x² = 0

=> y²/x² = (2y/x)•dy/dx

=> y = 2x•dy/dx , which is the required differntial equation .