Advertisements
Advertisements
प्रश्न
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Advertisements
उत्तर
\[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos \left( \frac{y}{x} \right) + x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y \cos \left( \frac{y}{x} \right) + x}{x \cos \left( \frac{y}{x} \right)}\]
\[\text { This is a homogeneous differential equation } . \]
\[\text { Putting }y = vx and \frac{dy}{dx} = v + x\frac{dv}{dx}, \text { we get }\]
\[v + x\frac{dv}{dx} = \frac{vx \cos v + x}{x \cos v}\]
\[\Rightarrow v + x\frac{dv}{dx} = \frac{v \cos v + 1}{\cos v}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{v \cos v + 1 - v \cos v}{\cos v}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{1}{\cos v}\]
\[ \Rightarrow \cos v dv = \frac{1}{x}dx\]
\[\text { Integrating both sides, we get }\]
\[\int\cos v \ dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \sin v = \log \left| x \right| + \log\left| C \right|\]
\[\text { Putting v }= \frac{y}{x}, we get\]
\[\sin\frac{y}{x} = \log \left| Cx \right|\]
\[\text { which is the general solution of the given differential equation } .\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 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.
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.`
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solution of the differential equation `"dx"/x + "dy"/y` = 0 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 `(x + 2y^3) "dy"/"dx"` = y
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
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 solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
