English

The function y = ex is solution ______ of differential equation

Advertisements
Advertisements

Question

The function y = ex is solution  ______ of differential equation

Fill in the Blanks
Advertisements

Solution

`("d"y)/("d"x) = y`

shaalaa.com
  Is there an error in this question or solution?
Chapter 1.8: Differential Equation and Applications - Q.2

APPEARS IN

RELATED QUESTIONS

\[x + \left( \frac{dy}{dx} \right) = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.


Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]

 


Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]


For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[y = \left( \frac{dy}{dx} \right)^2\]
\[y = \frac{1}{4} \left( x \pm a \right)^2\]

Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]

Function y = ex + 1


xy (y + 1) dy = (x2 + 1) dx


\[\frac{dy}{dx} = \frac{e^x \left( \sin^2 x + \sin 2x \right)}{y\left( 2 \log y + 1 \right)}\]

y ex/y dx = (xex/y + y) dy


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


Solve the following initial value problem:-

\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]


Solve the following initial value problem:-

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


The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?


The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).


Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.


Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.


y2 dx + (x2 − xy + y2) dy = 0


In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(a^2-x^2)`              `x+y(dy/dx)=0`


Form the differential equation representing the family of curves y = a sin (x + b), where ab are arbitrary constant.


Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.


Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.


Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.


Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2). 


Solve the following differential equation.

xdx + 2y dx = 0


Solve

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


 `dy/dx = log x`


Solve the following differential equation y log y = `(log  y - x) ("d"y)/("d"x)`


State whether the following statement is True or False:

The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x 


Solve the following differential equation 

sec2 x tan y dx + sec2 y tan x dy = 0

Solution: sec2 x tan y dx + sec2 y tan x dy = 0

∴ `(sec^2x)/tanx  "d"x + square` = 0

Integrating, we get

`square + int (sec^2y)/tany  "d"y` = log c

Each of these integral is of the type

`int ("f'"(x))/("f"(x))  "d"x` = log |f(x)| + log c

∴ the general solution is

`square + log |tan y|` = log c

∴ log |tan x . tan y| = log c

`square`

This is the general solution.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×