If for the differential equation y′=yx+ϕ(xy), the general solution | Filo
filo Logodropdown-logo

Class 12

Math

Calculus

Differential Equations

view icon695
like icon150

If for the differential equation y′=yx+ϕ(xy), the general solution is y=xlog|Cx|, then ϕ(x/y) is given by

  1. −x2/y2
  2. −y2/x2
  3. x2/y2
  4. −y2/x2
Correct Answer: Option(d)
Solution: [d] Putting v=y/x so that xdvdx+v=dvdx We have xdvdx+v=v+ϕ(1/v) ⇒dvϕ(1/v)=dxx;⇒log|Cx|=∫dvϕ(1/v) But y=xlog|Cx| is the general solution, So xy=1v=log|Cx|=∫dvϕ(1/v) ⇒ϕ(1/v)=−1/v2 (Differentiating w.r.t.v both sides) ⇒ϕ(x/y)=−y2/x2
view icon695
like icon150
filo banner image

Connecting you to a tutor in 60 seconds.

Get answers to your doubts.

playstore logoplaystore logo
Similar Topics
relations and functions ii
integrals
trigonometric functions
inverse trigonometric functions
application of derivatives