A homogeneous differential equation of the from dxdy=h(xy) can be solved by making the substitution. - Mathematics

Question

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

Options

y = vx
v = yx
x = vy
x = v

MCQ

Solution

x = vy

Explanation:

By substituting x = vy, where `v = x/y`, the differential equation `dx/dy = h (x/y)` is transformed into a separable from, making it easier to solve.

shaalaa.com

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

Is there an error in this question or solution?

Q 16 Q 15 Q 17

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

APPEARS IN

NCERT Mathematics [English] Class 12

Chapter 9 Differential Equations
Exercise 9.5 | Q 16 | Page 406

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

Show that the differential equation 2y^x^/y dx + (y − 2x e^x/^y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

Solve the differential equation :

`y+x dy/dx=x−y dy/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`

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

Which of the following is a homogeneous differential equation?

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.

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

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

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

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

(x² + 3xy + y²) dx − x² dy = 0

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

(2x² y + y³) dx + (xy² − 3x³) dy = 0

Solve the following initial value problem:
(x² + y²) dx = 2xy dy, y (1) = 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 (x² + 3y²) dx + y (y² + 3x²) dy = 0, y (1) = 1

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

Which of the following is a homogeneous differential equation?

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