Show that the given differential equation is homogeneous and solve them. x dy-y dx= x2+y2 dx - Mathematics

Question

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

`x dy - y dx = sqrt(x^2 + y^2) dx`

Sum

Solution

`x dy - y dx = sqrt(x^2 + y^2) dx`,

Which can be written as `x dy/dx = y + sqrt (x^2 + y^2)`

or `dy/dx = y/x + sqrt (1 + (y/x)^2)` ....(1)

Since R.H.S. is of the form `g(y/x)`, and so it is a homogeneous function of degree zero.

Therefore equation (1) is a homogeneous differential equation.

To solve this, put y = vx

⇒ `dy/dx = v + x (dv)/dx`

Substituting the value of y and `dy/dx` in (1), we get

`v + x (dv)/dx = v + sqrt (1 + v^2)`

⇒ `x (dv)/dx = sqrt(1 + v^2)`

⇒ `dx/x = (dv)/sqrt(1 + v^2)`

⇒ `int dx/x = int (dv)/ sqrt(1 + v^2)`

⇒ `log x + log C_1 = log |v + sqrt (1+ v^2)|`

⇒ `log x + log C_1 = log |y/x + sqrt (1 + y^2/x^2)|`

⇒ `log C_1 x = log |y + sqrt (x^2 + y^2)| - log x`

⇒ `pm C_1 x^2 = y + sqrt (x^2 + y^2)`

⇒ `Cx^2 = y + sqrt (x^2 + y^2)`

shaalaa.com

Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

Is there an error in this question or solution?

Q 6 Q 5 Q 7

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 6 | Page 406

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.

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.

`{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.

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

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

(x + y) dy + (x – y) dx = 0; y = 1 when x = 1

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:

`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

Which of the following is a homogeneous differential equation?

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

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

(x² + 3xy + y²) dx − x² dy = 0

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

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

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:
(y⁴ − 2x³ y) dx + (x⁴ − 2xy³) dy = 0, y (1) = 1

Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]

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 \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.

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 )`

Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.