Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Advertisements
उत्तर
y dx + (x - y2 ) dy = 0
∴ y dx = (y2 - x) dy
∴ `dx/dy = (y^2 - x) /y= y - x/y `
∴ `dx/dy + x/y = y`
The given equation is of the form
`dx/dy + Px = Q`
where, P = `1/y` and Q = y
∴ I.F. = `e int^ (pdy) = e int ^(1/ydy) = e ^(log |y|)= y`
∴ Solution of the given equation is
`x (I.F.) =int Q (I.F.) dy + c_1`
∴ `xy = int y(y) dy = y^3/3 + c_1`
∴ 3xy = y3 + c …[3c1 = c]
APPEARS IN
संबंधित प्रश्न
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
x cos y dy = (xex log x + ex) dx
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
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.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
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` |
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the differential equation:
`e^(dy/dx) = x`
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
Choose the correct alternative:
Differential equation of the function c + 4yx = 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
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log 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`
Solve the differential equation `"dy"/"dx" + 2xy` = y
