In each of the following examples, verify that the given function is a solution of the corresponding differential equation. Solution D.E. y = xn x2d2ydx2-n×xdydx+ny=0 - Mathematics and Statistics

Advertisements
Advertisements
Sum

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

Solution D.E.
y = xn `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0`
Advertisements

Solution

y = x n

Differentiating w.r.t. x, we get

`dy/dx = nx^(n-1)`

Again, differentiating w.r.t. x, we get

`(d^2y)/dx^2 = n(n-1) x^(n-2)`

∴  `x^2(d^2y)/dx^2 - nxdy/dx +ny`

= n(n-1)x2xn-2 - nx.nxn-1+ nxn

= n(n-1)xn - n2 xn + nxn

=[n(n-1)-n2+n]xn

= 0

∴ `x^2 (d^2y)/dx^2 - nxdy/dx + ny = 0`

∴ Given function is a solution of the given differential equation.

  Is there an error in this question or solution?
Chapter 8: Differential Equation and Applications - Exercise 8.1 [Page 162]

APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Differential Equation and Applications
Exercise 8.1 | Q 2.2 | Page 162

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`


Show that y = AeBx is a solution of the differential equation

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

Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\]  satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]


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

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

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

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

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

\[\frac{dy}{dx} = \tan^{- 1} x\]


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

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

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

xy (y + 1) dy = (x2 + 1) dx


\[5\frac{dy}{dx} = e^x y^4\]

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

(ey + 1) cos x dx + ey sin x dy = 0


xy dy = (y − 1) (x + 1) dx


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

\[\frac{dy}{dx} = 1 - x + y - xy\]

dy + (x + 1) (y + 1) dx = 0


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

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

 


Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]


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

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

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

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

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

Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]


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.


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

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

(x2 − y2) dx − 2xy dy = 0


\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]


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

Solve the following initial value problem:-

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


The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?


Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.


A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.


The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.


The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).


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 differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is


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


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


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 = 1 − logx `x^2(d^2y)/dx^2 = 1`

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

Solution D.E.
ax2 + by2 = 5 `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx`

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


Solve the following differential equation.

`y^3 - dy/dx = x dy/dx`


Solve the following differential equation.

y2 dx + (xy + x2 ) dy = 0


Solve the following differential equation.

y dx + (x - y2 ) dy = 0


Solve the following differential equation.

`dy/dx + 2xy = x`


Choose the correct alternative.

The differential equation of y = `k_1 + k_2/x` is


Choose the correct alternative.

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


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


Choose the correct alternative.

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


Solve

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


`xy dy/dx  = x^2 + 2y^2`


 `dy/dx = log x`


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


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


The function y = ex is solution  ______ of differential equation


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 `("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

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


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.


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×