Advertisements
Advertisements
प्रश्न
`xy dy/dx = x^2 + 2y^2`
Advertisements
उत्तर
`xy dy/dx = x^2 + 2y^2`
∴ `dy/dx = x^2 + (2y^2)/ (xy)` …(i)
Put y = tx ...(ii)
Differentiating w.r.t. x, we get
`dy/dx = t + x dt/dx` ...(iii)
Substituting (ii) and (iii) in (i), we get
`t + x dt/dx = (x^2 + 2t^2 x^2)/ (x(tx))`
∴ `t + x dt/dx = (x^2 (1+2t^2))/(x^2t)`
∴ `x dt/dx (1+2t^2)/t - t = (1+ t^2)/t`
∴ `t/(1+t^2) dt = 1/x dx`
Integrating on both sides, we get
`1/2 int (2t)/(1+t^2) dt = int dx/x`
∴ `1/2 log|1 + t^2| = log |x| + log |c|`
∴ log |1 + t2 | = 2 log |x| + 2log |c|
= log |x2 | + log |c2|
∴ log |1 + t2 | = log |c2 x2|
∴ 1 + t2 = c2x2
∴ `1 + y^2/x^2 = c^2x^2`
∴ x2 + y2 = c2 x4
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
(ey + 1) cos x dx + ey sin x dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
x2 dy + y (x + y) dx = 0
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Form the differential equation from the relation x2 + 4y2 = 4b2
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Solve the differential equation:
`e^(dy/dx) = x`
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the differential equation xdx + 2ydy = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
