Advertisements
Advertisements
प्रश्न
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Advertisements
उत्तर
We have,
\[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y \cos\left( \frac{y}{x} \right) + x}{x \cos \left( \frac{y}{x} \right)} . . . . . \left( 1 \right)\]
Clearly this is a homogeneous equation,
Putting y = vx
\[ \Rightarrow \frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[\text{Substituting }y = vx\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx}\text{ in (1) we get}\]
\[\frac{dy}{dx} = \frac{y \cos\left( \frac{y}{x} \right) + x}{x \cos \left( \frac{y}{x} \right)}\]
\[ \Rightarrow v + x\frac{dv}{dx} = \frac{vx \cos \left( v \right) + x}{x \cos \left( v \right)}\]
\[ \Rightarrow v + x\frac{dv}{dx} = \frac{v \cos \left( v \right) + 1}{\cos \left( v \right)}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{v \cos \left( v \right) + 1}{\cos \left( v \right)} - v\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{v \cos \left( v \right) + 1 - v \cos \left( v \right)}{\cos \left( v \right)}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{1}{\cos \left( v \right)}\]
\[ \Rightarrow \cos \left( v \right) dv = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int\cos \left( v \right) dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \sin \left( v \right) = \log \left| x \right| + \log \left| C \right|\]
\[ \Rightarrow \sin \left( \frac{y}{x} \right) = \log \left| Cx \right|\]
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.
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
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 \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Find the differential equation of all non-horizontal lines in a plane.
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The solution of differential equation coty dx = xdy is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
