Show that the given differential equation is homogeneous and solve them. xdydx -y+ xsin(yx)=0 - Mathematics

Question

Show that the given differential equation is homogeneous and solve them.

`x dy/dx - y + x sin (y/x) = 0`

Sum

Solution

`x dy/dx - y + x sin (y/x) = 0`

`dy/dx = y/x - sin y/x = g (y/x)` (say) ....(i)

The right side of the equation is in the form of `g(y/x)` so it is a homogeneous differential equation of zero degree.

∴Putting y = vx

`dy/dx = v + x dy/dx` From equation (i)

`=> v + x (dv)/dx = v - sin v`

`=> x (dv)/dx = v - sin v - v`

`=> x (dv)/dx = - sin v`

`=> cosec v dv = - 1/x dx`

⇒ log |cosec v - cot v|

= - log |x| + C₁

On integrating,

⇒ log |(cosec v - cot v)| = C₁

⇒ |x (cosec v - cot v)| = e^C₁

⇒ x (cosec v - cot v) = ± e^C₁ = C (say)

⇒ `x (cosec y/x - cot y/x) = C`

Which is the required general solution.

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Homogeneous Differential Equations

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Q 8 Q 7 Q 9

Chapter 9: Differential Equations - Exercise 9.5 [Page 406]

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NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 9 Differential Equations
Exercise 9.5 | Q 8 | Page 406

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