A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution - Mathematics and Statistics

Advertisements
Advertisements
Fill in the Blanks

A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution

Advertisements

Solution

particular

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

RELATED QUESTIONS

Prove that :

`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`


If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega +  b omega^2) =  omega^2`


Solve the equation for x: `sin^(-1)  5/x + sin^(-1)  12/x = pi/2, x != 0`


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

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


Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].


Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]


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

Differential equation Function
\[x\frac{dy}{dx} + y = y^2\]
\[y = \frac{a}{x + a}\]

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


\[\frac{dy}{dx} = x^5 \tan^{- 1} \left( x^3 \right)\]

\[\sin^4 x\frac{dy}{dx} = \cos x\]

\[\sqrt{a + x} dy + x\ dx = 0\]

\[\frac{dy}{dx} = x \log x\]

\[\left( x^3 + x^2 + x + 1 \right)\frac{dy}{dx} = 2 x^2 + x\]

Solve the differential equation \[\frac{dy}{dx} = e^{x + y} + x^2 e^y\].

(ey + 1) cos x dx + ey sin x dy = 0


\[\frac{dy}{dx} = e^{x + y} + e^y x^3\]

(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0


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

Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.


In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).


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

\[\cos^2 \left( x - 2y \right) = 1 - 2\frac{dy}{dx}\]

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

x2 dy + y (x + y) dx = 0


\[\frac{dy}{dx} = \frac{y - x}{y + x}\]

(x2 − y2) dx − 2xy dy = 0


3x2 dy = (3xy + y2) dx


(x + 2y) dx − (2x − y) dy = 0


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

 

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]


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?


A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.


Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.


The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.


Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\]  are rectangular hyperbola.


Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]


The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is


The solution of the differential equation y1 y3 = y22 is


Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]


The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution


If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]


Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]


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`


If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`


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


Solve the differential equation:

`"x"("dy")/("dx")+"y"=3"x"^2-2`


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). 


The price of six different commodities for years 2009 and year 2011 are as follows: 

Commodities A B C D E F

Price in 2009 (₹)

35 80 25 30 80 x
Price in 2011 (₹) 50 y 45 70 120 105

The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.


Choose the correct option from the given alternatives:

The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of


Choose the correct option from the given alternatives:

The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is


Determine the order and degree of the following differential equations.

Solution D.E
y = aex + be−x `(d^2y)/dx^2= 1`

Form the differential equation from the relation x2 + 4y2 = 4b2


For each of the following differential equations find the particular solution.

(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0


For each of the following differential equations find the particular solution.

`y (1 + logx)dx/dy - x log x = 0`,

when x=e, y = e2.


Solve the following differential equation.

(x2 − y2 ) dx + 2xy dy = 0


Solve the following differential equation.

`dy/dx + y` = 3


Solve the following differential equation.

y dx + (x - y2 ) dy = 0


Solve the following differential equation.

dr + (2r)dθ= 8dθ


Choose the correct alternative.

The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.


The solution of `dy/ dx` = 1 is ______


The solution of `dy/dx + x^2/y^2 = 0` is ______


A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.


State whether the following is True or False:

The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.


Select and write the correct alternative from the given option for the question

Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in


Select and write the correct alternative from the given option for the question

The differential equation of y = Ae5x + Be–5x is


Select and write the correct alternative from the given option for the question 

Differential equation of the function c + 4yx = 0 is


Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0


Solve the differential equation `("d"y)/("d"x) + y` = e−x 


Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`


Solve the differential equation xdx + 2ydy = 0


Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0


Solve the following differential equation `("d"y)/("d"x)` = x2y + y


Solve: `("d"y)/("d"x) + 2/xy` = x2 


For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0


Solve the following differential equation

`yx ("d"y)/("d"x)` = x2 + 2y2 


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


For the differential equation, find the particular solution

`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0


Solve the following differential equation y2dx + (xy + x2) dy = 0


Solve the following differential equation

`x^2  ("d"y)/("d"x)` = x2 + xy − y2 


Choose the correct alternative:

Differential equation of the function c + 4yx = 0 is


Choose the correct alternative:

General solution of `y - x ("d"y)/("d"x)` = 0 is


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.


Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)

Solution: `("d"y)/("d"x)` = cos(x + y)    ......(1)

Put `square`

∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`

∴ (1) becomes `"dv"/("d"x) - 1` = cos v

∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

Integrating, we get

`int 1/(1 + cos "v")  "d"v = int  "d"x`

∴ `int 1/(2cos^2 ("v"/2))  "dv" = int  "d"x`

∴ `1/2 int square  "dv" = int  "d"x`

∴ `1/2* (tan("v"/2))/(1/2)` = x + c

∴ `square` = x + c


The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0


lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is


There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?


If `y = log_2 log_2(x)` then `(dy)/(dx)` =


A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is


`d/(dx)(tan^-1  (sqrt(1 + x^2) - 1)/x)` is equal to:


The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:


Solve the differential equation

`y (dy)/(dx) + x` = 0


Solve the differential equation

`x + y dy/dx` = x2 + y2


Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.


Share
Notifications



      Forgot password?
Use app×