Advertisements
Advertisements
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x^{2} + xy − y^{2}
Advertisements
Solution
`x^2 ("d"y)/("d"x)` = x^{2} + xy − y^{2}
∴ `("d"y)/("d"x) = 1 + y/x - (y/x)^2` .....(i)
Put `y/x` = t .....(ii)
∴ y = tx
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "t" + x ("dt")/("d"x)` .....(iii)
Substituting (ii) and (iii) in (i), we get
`"t" + x "dt"/("d"x)` = 1 + t − t^{2}
∴ `x "dt"/("d"x)` = 1 − t^{2}
∴ `"dt"/(1 - "t"^2) = ("d"x)/x`
Integrating on both sides, we get
`int "dt"/(1 - "t"^2) = int ("d"x)/x`
∴ `1/2 log|(1 + t)/(1 - t)|` = log |x| + log |c_{1}|
∴ `log |(1 + y/x)/(1 - y/x)|` = 2log |x| + 2log |c_{1}|
∴ `log|(x + y)/(x - y)|` = log |x^{2}| + log |c_{1}^{2}|
∴ `log|(x + y)/(x - y)|` = log |c^{1}x^{2}|
∴ `(x + y)/(x - y)` = c_{1}^{2}x^{2}
∴ `(x + y)/(x - y)` = cx^{2}, where c = c_{1}^{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`
Show that y = Ae^{Bx} is a solution of the differential equation
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
x cos y dy = (xe^{x} log x + e^{x}) dx
(e^{y} + 1) cos x dx + e^{y} sin x dy = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
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:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
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.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The differential equation satisfied by ax^{2} + by^{2} = 1 is
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
y^{2} dx + (x^{2} − xy + y^{2}) dy = 0
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
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 x^{2} – 3y^{2} – 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 = ae^{x }+ be^{−x} | `(d^2y)/dx^2= 1` |
Find the differential equation whose general solution is
x^{3} + y^{3} = 35ax.
Form the differential equation from the relation x^{2 }+ 4y^{2 }= 4b^{2}
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
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.
Solve the differential equation:
`e^(dy/dx) = x`
x^{2}y dx – (x^{3} + y^{3}) dy = 0
`dy/dx = log x`
y dx – x dy + log x dx = 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 = Ae^{5x} + 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 sec^{2}y tan x dy + sec^{2}x 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 (x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
Solve: `("d"y)/("d"x) + 2/xy` = x^{2}
For the differential equation, find the particular solution (x – y^{2}x) dx – (y + x^{2}y) dy = 0 when x = 2, y = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x^{2} + 2y^{2}
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 y^{2}dx + (xy + x^{2}) dy = 0
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 solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 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}
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)` = x^{2}y + 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 `("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
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 + y^{2})x dx – (1 + x^{2})y dy = 0 represents a family of:
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x^{2} + y^{2}
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.