Advertisements
Advertisements
प्रश्न
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Advertisements
उत्तर
Given that: (1 + tan y)(dx – dy) + 2xdy = 0
⇒ (1 + tan y)dx – (1 + tan y)dy + 2xdy = 0
⇒ (1 + tan y)dx – (1 + tan y – 2x)dy = 0
⇒ `(1 + tan y) "dx"/"dy" = (1 + tan y - 2x)`
⇒ `"dx"/"dy" = (1 + tan y - 2x)/(1 + tan y)`
⇒ `"dx"/"dy" = 1 - (2x)/(1 + tan y)`
⇒ `"dx"/"dy" + (2x)/(1 + tan y)` = 1
Here, P = `2/(1 + tan y)` and Q = 1
Integrating factor I.F.
= `"e"^(int 2/(1 + tan y) "dy")`
= `"e"^(int (2cosy)/(siny + cosy)"d"y)`
= `"e"^(int (siny + cosy - siny + cosy)/((siny + cosy)) "dy"`
= `"e"^(int(1 + (cosy - siny)/(siny + cosy))"d"y)`
= `"e"^(int 1."d"y) . "e"^(int(cosy - siny)/(siny + cosy)"d"y)`
= `"e"^y . "e"^(log(siny + cosy)`
= `"e"^y . (siny + cos y)`
So, the solution is `x xx "I"."F". = int "Q" xx "I"."F". "d"y + "c"`
⇒ `x . "e"^y (siny + cosy) = int 1 . "e"^y (siny + cosy)"d"y + "c"`
⇒ `x . "e"^y )siny + cosy) = "e"^y . sin y + "c"` .....`[because int x^x "f"(x) + "f'"(x)]"d"x = "e"^x "f"(x) + "c"]`
⇒ `x(siny + cos y) = sin y + "c" . "e"^-y`
Hence, the required solution is `x(siny + cos y) = sin y + "c" . "e"^-y`.
APPEARS IN
संबंधित प्रश्न
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
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)
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
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.`
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 solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
(x3 − 2y3) dx + 3x2 y dy = 0
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
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 equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
Which of the following differential equations has `y = x` as one of its particular solution?
