Solve: dydx+2xy = x2 - Mathematics and Statistics

Advertisements
Advertisements
Sum

Solve: `("d"y)/("d"x) + 2/xy` = x2 

Advertisements

Solution

`("d"y)/("d"x) + 2/xy` = x2 

The given equation is of the form

`("d"y)/("d"x) + "P"y` = Q.

where P = `2/x` and Q = x2

∴ I.F. = `"e"^(int^("Pd"x))`

= `"e"^(int2/x)  "d"x`

= `"e"^(2logx)`

= `"e"^(log x^2)`

= x2

∴ Solution of the given equation is

`y("I"."F".) = int"Q"("I.""F.")  "d"x + "c"_1`

∴ `y * x^2 = intx^2 * x^2  "d"x + "c"_1`

∴ `yx^2 = intx^4  "d"x + "c"_1`

∴ yx2 = `x^5/5 + "c"_1`

∴ 5x2y = x5 + c, where c = 5c1 

  Is there an error in this question or solution?
Chapter 1.8: Differential Equation and Applications - Q.4

Video TutorialsVIEW ALL [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`


\[\frac{d^4 y}{d x^4} = \left\{ c + \left( \frac{dy}{dx} \right)^2 \right\}^{3/2}\]

\[y\frac{d^2 x}{d y^2} = y^2 + 1\]

Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.

 

Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]


Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]


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

 


Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].


Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.


\[\frac{dy}{dx} = x^2 + x - \frac{1}{x}, x \neq 0\]

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

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

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

\[\sqrt{1 - x^4} dy = x\ dx\]

\[\frac{dy}{dx} = x e^x - \frac{5}{2} + \cos^2 x\]

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

\[\frac{dy}{dx} + \frac{1 + y^2}{y} = 0\]

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

\[x\frac{dy}{dx} + \cot y = 0\]

dy + (x + 1) (y + 1) dx = 0


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

Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]

 


\[\frac{dy}{dx} = 2xy, y\left( 0 \right) = 1\]

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

The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.


Solve the following initial value problem:-

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


Solve the following initial value problem:-

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


Solve the following initial value problem:-

\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]


Solve the following initial value problem:-

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


Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]


The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.


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.


The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.


Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\]  = x (x + 1) and passing through (1, 0).


The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).


The solution of the differential equation y1 y3 = y22 is


Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is


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


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


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


In each of the following examples, verify that the given function is a solution of the corresponding differential equation.

Solution D.E.
xy = log y +k y' (1-xy) =y2

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.

xdx + 2y dy = 0


Solve the following differential equation.

(x2 − y2 ) dx + 2xy dy = 0


Solve the following differential equation.

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


Solve the following differential equation.

`dy/dx + y = e ^-x`


Solve the following differential equation.

dr + (2r)dθ= 8dθ


Choose the correct alternative.

The solution of `x dy/dx = y` log y is


Choose the correct alternative.

The integrating factor of `dy/dx -  y = e^x `is ex, then its solution 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.


y2 dx + (xy + x2)dy = 0


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


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:

Solution of the equation `x("d"y)/("d"x)` = y log y is


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


The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`


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


Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0


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


Solve: ydx – xdy = x2ydx.


Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]


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×