Advertisements
Advertisements
प्रश्न
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Advertisements
उत्तर
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
∴ `("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 − t2
∴ `x "dt"/("d"x)` = 1 − t2
∴ `"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 |c1|
∴ `log |(1 + y/x)/(1 - y/x)|` = 2log |x| + 2log |c1|
∴ `log|(x + y)/(x - y)|` = log |x2| + log |c12|
∴ `log|(x + y)/(x - y)|` = log |c1x2|
∴ `(x + y)/(x - y)` = c12x2
∴ `(x + y)/(x - y)` = cx2, where c = c12
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
dy + (x + 1) (y + 1) dx = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
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:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
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.
y ex/y dx = (xex/y + y) dy
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
(y2 − 2xy) dx = (x2 − 2xy) dy
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
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.
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 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.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
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 parabolas having vertex at origin and axis along positive direction of x-axis.
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 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` = 3
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve the following differential equation.
`dy/dx + 2xy = x`
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.
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
`xy dy/dx = x^2 + 2y^2`
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
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 ______
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.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
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)` =
