Advertisements
Advertisements
प्रश्न
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Advertisements
उत्तर
`yx ("d"y)/("d"x)` = x2 + 2y2
∴ `("d"y)/("d"x) = (x^2 + 2y^2)/(xy)` ......(i)
Put y = vx ......(ii)
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "v" + x "dv"/("d"x)` ......(iii)
Substituting (ii) and (iii) in (i), we get
`"v" + x "dv"/("d"x) = (x^2 + 2"v"^2x^2)/(x("v"x))`
∴ `"v" + x "dv"/("d"x) = (x^2(1 + 2"v"^2))/(x^2"v")`
∴ `x "dv"/("d"x) = (1 + 2"v"^2)/"v" - "v"`
= `(1 + "v"^2)/"v"`
∴ `"v"/(1 + "v"^2) "dv" = 1/x "d"x`
Integrating on both sides, we get
`1/ int (2"v")/(1 +"v"^2) "dv" = int "dv"/x`
∴ `1/2 log|1 + "v"^2|` = log |x| + log |c|
∴ log |1 + c2| = 2 og |x| + 2log |c|
= log |x2| + log |c2|
∴ log |1 + v2| = log |c2x2|
∴ 1 + v2 = c2x2
∴ `1 + y^2/x^2` = c2x2
∴ x2 + y2 = c2x4
APPEARS IN
संबंधित प्रश्न
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
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 \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(1 − x2) dy + xy dx = xy2 dx
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
dy + (x + 1) (y + 1) dx = 0
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
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.
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
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.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
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 particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
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 = aex + be−x | `(d^2y)/dx^2= 1` |
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
The differential equation of `y = k_1e^x+ k_2 e^-x` 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.
`dy/dx = log x`
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve the differential equation xdx + 2ydy = 0
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c1
∴ e–2y = – 2sin x – 2c1
∴ `square` = c, where c = – 2c1
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve: ydx – xdy = x2ydx.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
