Show that the given differential equation is homogeneous and solve them. x2dydx =x2-2y2+xy

Question

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

`x^2 dy/dx = x^2 - 2y^2 + xy`

Sum

Solution

Given `x^2 dy/dx = x^2 - 2y^2 + xy`

⇒ `dy/dx = (x^2 - 2y^2 + xy)/x^2`

`= 1 - 2 (y^2/x) + y/x` ....(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.

∴ put y = vx

⇒ `dy/dx = v.1 + x (dv)/dx`

Substituting these values of y and `dy/dx` in the given equation, we get

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

⇒ `x (dv)/dx = 1 - 2v^2`

⇒ `1/x dx = 1/ (1 - 2v^2) dv`

On integration, we get

`log |x| = int 1/ (1 - 2v^2) dv + C`

⇒ `log |x| = 1/2 int (dv)/ ((1/ sqrt2)^2 - v^2) + C`

⇒ `log |x| = 1/2 * 1/ (2* (1/sqrt2)) log |((1/sqrt2)+v)/((1/sqrt2) - v)| + C`

⇒ `log |x| = 1/(2sqrt2) log |(1 + sqrt(2v))/(1 - sqrt(2v))| + C`

⇒ `log |x| = 1/ (2 sqrt2) log |(1 + sqrt2* (y/x))/(1 - sqrt2 * (y/x))| + C`

⇒ `log |x| = 1/ (2sqrt2) log | (x + sqrt(2y))/(x - sqrt(2y))| + C`

Where C is an arbitrary constant

shaalaa.com

Forms of Solving Differential Equations> Homogeneous Differential Equations

Is there an error in this question or solution?

Q 5 Q 4 Q 6

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 5 | 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.

(x² + xy) dy = (x² + y²) dx

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.

`y dx + x log(y/x)dy - 2x dy = 0`

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 differential equation (x³ – 3x y²) dx = (y³ – 3x²y) dy, where c is a parameter.

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

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

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

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

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

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:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]

Solve the following initial value problem:
x (x² + 3y²) dx + y (y² + 3x²) dy = 0, y (1) = 1

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

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