Advertisements
Advertisements
प्रश्न
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
Advertisements
उत्तर
Here `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2d"y" = 0`
⇒ `(d"y")/(d"x") = ("x"^2 + 3"xy"+"y"^2)/("x"^2)`
⇒ `(d"y")/(d"x") = 1 + 3("y")/("x") + ("y"^2)/("x"^2)`
Put y = vx
⇒ `(d"y")/(d"x") = "v" + "x"(d"v")/(d"x")`
∴ `"v" + "x"(d"v")/(d"x") = 1 + 3"v"+"v"^2`
⇒ `"x"(d"v")/(d"x") = 1 + 2"v"+"v"^2`
⇒ `int_ (d"v")/(("v"+1)^2) = int_ (d"x")/("x")`
⇒ `-(1)/(("v"+1)) = log|"x"| + "C"`
⇒ `-("x")/("y"+"x") = log|"x"|+"C"`
As 1 y = 0 when x = 1 so, `-(1)/(0+1) = log|1|+"C"`
⇒ `"C" = -1`.
Hence the required solution is, `-("x")/("y"+"x") = log|"x"|-1`
⇒ `-"x" = "y" log|"x"| +"x" log |"x"| -"y" -"x"`
∴ `"y" = log|"x"| ("x" + "y")`
or,
`"y" = ("x" log|"x"|)/(1-log|"x"|)`.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
Which of the following differential equations has y = x as one of its particular solution?
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Solution of differential equation xdy – ydx = 0 represents : ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
