Advertisements
Advertisements
प्रश्न
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
Advertisements
उत्तर
Given differential equation is `x (dy)/(dx) = y(logy - logx + 1)`
⇒ `(dy)/(dx) = y/x(log y/x + 1)`
Put y = vx
⇒ `(dy)/(dx) = v + x(dv)/(dx)`
⇒ `v + x (dv)/(dx) = v(logv + 1)`
⇒ `(dv)/(vlogv) = (dx)/x`
On integrating both sides, we get
`int (dx)/x = int(dx)/x`
⇒ log(logv) = logx + logC
⇒ log(logv) = logCx
⇒ log(y/x) = Cx
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Solve the differential equation `dy/dx -y =e^x`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
(x2 + 1) dy + (2y − 1) dx = 0
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
