Home
Class 12
MATHS
The second derivative of a single valued...

The second derivative of a single valued function parametrically represented by `x=varphi(t)a n dy=psi(t)` (where `varphi(t)a n dpsi(t)` are different function and `varphi^(prime)(t)!=0` ) is given by `(d^2y)/(dx^2)=(((dx)/(dt))((d^2y)/(dt^2))-((d^2x)/(dt^2))((dy)/(dt))/(((dx)/(dt))^2)` `(d^2y)/(dx^2)=(((dx)/(dt))((d^2y)/(dt^2))-((d^2x)/(dt^2))((dy)/(dt))/(((dx)/(dt))^3)` `(d^2y)/(dx^2)=(((d^2x)/(dt))((d^y)/(dt^2))-((d^x)/(dt^))((d^2y)/(dt^2))/(((dx)/(dt))^3)` `(d^2y)/(dx^2)=(((d^2x)/(dt^2))((d^y)/(dt^))-((d^2x)/(dt^2))((d^y)/(dt^))/(((dx)/(dt))^3)`

A

`(d^(2)y)/(dx^(2))=(((dx)/(dt))((d^(2)y)/(dt^(2)))-((d^(2)x)/(dt^(2)))((dy)/(dt)))/(((dx)/(dt))^(3))`

B

`(d^(2)y)/(dx^(2))=(((dx)/(dt))((d^(2)y)/(dt^(2)))-((d^(2)x)/(dt^(2)))((dy)/(dt)))/(((dx)/(dt))^(2))`

C

`(d^(2)y)/(dx^(2))=(((d^(2)x)/(dt))((dy)/(dt))-(dx)/(dt)((d^(2)y)/(dt^(2))))/(((dx)/(dt))^(3))`

D

`(d^(2)y)/(dx^(2))=(((d^(2)x)/(dt))((dy)/(dt))-((d^(2)y)/(dt^(2)))((dy)/(dt)))/(((dy)/(dt))^(3))`

Text Solution

Verified by Experts

The correct Answer is:
A
`(dy)/(dx)=(dy//dt)/(dx//dt)`
`(d^(2)y)/(dx^(2))=(d)/(dx)((dy//dt)/(dx//dt))`
`=(((dx)/(dt))(d)/(dx)((dy)/(dt))-((dt)/(dt))(d)/(dx)((dx)/(dt)))/(((dx)/(dt))^(2))`
`=(((dx)/(dt))(d)/(dt)((dy)/(dt))(dt)/(dx)-((dy)/(dt))(d)/(dt)((dx)/(dt))(dt)/(dx))/(((dx)/(dt))^(2))`
`=(((dx)/(dt))(d^(2)y)/(dt^(2))-((dy)/(dt))(d^(2)x)/(dt^(2)))/(((dx)/(dt))^(3))`
Promotional Banner

Topper's Solved these Questions

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

  • MATRICES

    CENGAGE|Exercise Multiple Correct Answer|7 Videos

  • METHODS OF DIFFERETIATION

    CENGAGE|Exercise Question Bank|13 Videos

Similar Questions

Explore conceptually related problems

Solve (d^2y)/(dx^2)=((dy)/(dx))^2

If e^y(x+1)=1 , show that (d^2y)/(dx^2)=((dy)/(dx))^2

If x=logp and y=1/p ,then (a) (d^2y)/(dx^2)-2p=0 (b) (d^2y)/(dx^2)+y=0 (c) (d^2y)/(dx^2)+(dy)/(dx)=0 (d) (d^2y)/(dx^2)-(dy)/(dx)=0

If R=([1+((dy)/(dx))^2]^(3//2))/((d^2y)/(dx2)) , then R^(2//3) can be put in the form of 1/(((d^2y)/(dx^2))^(2//3))+1/(((d^2x)/(dy^2))^(2//3)) b. 1/(((d^2y)/(dx^2))^(2//3))-1/(((d^2x)/(dy^2))^(2//3)) c. 2/(((d^2y)/(dx^2))^(2//3))+2/(((d^2x)/(dy^2))^(2//3)) d. 1/(((d^2y)/(dx^2))^(2//3))dot1/(((d^2x)/(dy^2))^(2//3))

In which of the following differential equation degree is not defined? (a) (d^2y)/(dx^2)+3(dy/dx)^2=xlog((d^2y)/(dx^2)) (b) ((d^2y)/(dx^2))^2+(dy/dx)^2=xsin((d^2y)/(dx^2)) (c) x=sin((dy/dx)-2y),|x|<1 (d) x-2y=log(dy/dx)

(a+bx)e^(y/x)=x , Prove that x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^2

Find the order and degree of the following differential equations. i) (dy)/(dx)+y=1/((dy)/(dx)) , ii) e^((d^(3)y)/(dx^(3)))-x(d^(2)y)/(dx^(2))+y=0 , iii) sin^(-1)((dy)/(dx))=x+y , iv) log_(e)((dy)/(dx))=ax+by v) y(d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)-4y(dy)/(dx)=0

If y=xlog(x/(a+b x)),t h e nx^3(d^2y)/(dx^2)= (a) (x(dy)/(dx)-y) ^2 (b) x (dy/dx)-y (c) y(dy/dx)-x (d) (y(dy/dx)-x)^2

If y^2=P(x) , then 2d/dx(y^3(d^2y/dx^2))