Advertisements
Advertisements
प्रश्न
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Advertisements
उत्तर
We have,
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
\[\frac{dy}{dx} + \cos^2 \left( \frac{y}{x} \right) = \frac{y}{x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x} - \cos^2 \left( \frac{y}{x} \right)\]
Putting `y = vx,` we get
\[\frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[ \therefore v + x\frac{dv}{dx} = v - \cos^2 \left( v \right)\]
\[ \Rightarrow x\frac{dv}{dx} = - \cos^2 v\]
\[ \Rightarrow \sec^2 v\ dv = - \frac{1}{x}dx\]
Integrating both sides, we get
\[\int sec^2 v\ dv = - \int\frac{1}{x}dx\]
\[ \Rightarrow \tan v = - \log x + C\]
\[ \Rightarrow \tan \frac{y}{x} = - \log x + C\]
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
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 = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2 y′ + y2 + 1 = 0
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
If y = etan x+ (log x)tan x then find dy/dx
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
x2 dy + (x2 − xy + y2) dx = 0
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
