English

A Homogeneous Differential Equation of the Form D X D Y = H ( X Y ) Can Be Solved by Making the Substitution

Advertisements
Advertisements

Question

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution

Options

  • y = vx

  • v = yx

  • x = vy

  • x = v

MCQ
Advertisements

Solution

x = vy

 

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\]  can be solved by substituting x = vy.

shaalaa.com
  Is there an error in this question or solution?
Chapter 21: Differential Equations - MCQ [Page 143]

APPEARS IN

R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 21 Differential Equations
MCQ | Q 48 | Page 143

RELATED QUESTIONS

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.

`x^2 dy/dx = x^2 - 2y^2 + xy`


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:

(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


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.


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

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

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

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

(x2 + 3xy + y2) dx − x2 dy = 0


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

Solve the following initial value problem:
 (x2 + y2) dx = 2xy dy, y (1) = 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:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]


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


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


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.


Solve the following differential equation:

y2 dx + (xy + x2)dy = 0


Solve the following differential equation:

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


Solve the following differential equation:

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


Solve the following differential equation:

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


Solve the following differential equation:

`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`


Solve the following differential equation:

(x2 + 3xy + y2)dx - x2 dy = 0


Solve the following differential equation:

(x2 – y2)dx + 2xy dy = 0


The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is


If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy = (2xy + y2)dx, then `f(1/2)` is equal to ______.


Which differential equation is called a homogeneous differential equation?


A function \[F(x,y)\] is homogeneous of degree \[n\] when which condition holds?


For the substitution \[y=vx\], which differentiated form is correct?


Which substitutions are chosen for a homogeneous differential equation?


What replacement obtains the final answer after using a homogeneous substitution?


Which final answer results after replacing \[v=\frac{y}{x}\] in \[\sin v=\ln|Cx|\]?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×