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

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.

shaalaa.com

Methods of Solving Differential Equations> Homogeneous Differential Equations

Is there an error in this question or solution?

Q 8 Q 7 Q 9

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

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12

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

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Find the particular solution of the differential equation:

2y e^x/y dx + (y - 2x e^x/y) dy = 0 given that x = 0 when y = 1.

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

(x² – y²) dx + 2xy dy = 0

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

`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`

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

`(1+e^(x/y))dx + e^(x/y) (1 - x/y)dy = 0`

For the differential equation find a particular solution satisfying the given condition:

x² dy + (xy + y²) dx = 0; y = 1 when x = 1

For the differential equation find a particular solution satisfying the given condition:

`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "when" x = 1`

For the differential equation find a particular solution satisfying the given condition:

`dy/dx - y/x + cosec (y/x) = 0; y = 0` when x = 1

For the differential equation find a particular solution satisfying the given condition:

`2xy + y^2 - 2x^2 dy/dx = 0; y = 2` when x = 1

A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.

Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.

\[\frac{y}{x}\cos\left( \frac{y}{x} \right) dx - \left\{ \frac{x}{y}\sin\left( \frac{y}{x} \right) + \cos\left( \frac{y}{x} \right) \right\} dy = 0\]

\[\left( 1 + e^{x/y} \right) dx + e^{x/y} \left( 1 - \frac{x}{y} \right) dy = 0\]

\[\left( x^2 + y^2 \right)\frac{dy}{dx} = 8 x^2 - 3xy + 2 y^2\]

(x² − 2xy) dy + (x² − 3xy + 2y²) dx = 0

(x² + 3xy + y²) dx − x² dy = 0

\[\left( x - y \right)\frac{dy}{dx} = x + 2y\]

\[y dx + \left\{ x \log\left( \frac{y}{x} \right) \right\} dy - 2x dy = 0\]

Solve the following initial value problem:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]

Solve the following initial value problem:
(xy − y²) dx − x² dy = 0, y(1) = 1

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]

Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]

Solve the following differential equation : \[\left[ y - x \cos\left( \frac{y}{x} \right) \right]dy + \left[ y \cos\left( \frac{y}{x} \right) - 2x \sin\left( \frac{y}{x} \right) \right]dx = 0\] .

Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`