Show that the given differential equation is homogeneous and solve them. (x – y) dy – (x + y) dx = 0 - Mathematics

Question

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

(x – y) dy – (x + y) dx = 0

Sum

Solution

(x - y) dy - (x + y) dx = 0

`=> dy/dx = (x + y)/(x - y)`

`= (1 + (y/x))/(1 - (y/x))`

∵ The powers of the numerator and denominator are the same so this is a homogeneous differential equation.

∴ Putting y = vx

From equation (i),

`dy/dx = v + x (dv)/dx`

`v + x (dv)/dx = (x + vx)/(x - vx)`

`=> x (dv)/dx = (1 + v)/(1 - v) - v`

`=> x dy/dx= (1 + v - v + v^2)/(1 - v)`

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

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

Integrating on both sides

`=> int ((1 - v)/(1 + v^2)) dv = 1/x dx`

`=> int 1/(v^2 + 1) v - 1/2 int (2v)/(v^2 + 1) dv = int 1/x dx`

`=> tan^-1 v = 1/2 log (v^2 + 1) + log x + C`

`=> tan^-1 (y/x) = 1/2 log (y^2/x^2 + 1) + log x + C because y = vx`

`=> tan^-1 (y/x) = 1/2 log ((y^2 + x^2)/x^2) + log x + C`

`=> tan^-1 (y/x) = 1/2 log (x^2 + y^2) - 1/2 log x^2 + log x + C`

`=> tan^-1 (y/x) = 1/2 log (x^2 + y^2) + C`

shaalaa.com

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

Is there an error in this question or solution?

Q 3 Q 2 Q 4

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

APPEARS IN

NCERT Mathematics [English] Class 12

Chapter 9 Differential Equations
Exercise 9.5 | Q 3 | Page 406

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

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

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

(x² + xy) dy = (x² + y²) dx

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`

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`

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

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²)dx = (y³ – 3x²y)dy, where C is parameter

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

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

\[x\frac{dy}{dx} - y = 2\sqrt{y^2 - x^2}\]

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

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

(2x² y + y³) dx + (xy² − 3x³) dy = 0

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

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

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

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 )`