Advertisements
Advertisements
प्रश्न
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} = \frac{y}{x} + \sin \left( \frac{y}{x} \right)\]
This is a homogeneous differential equation.
\[\text{Putting }y = vx\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx},\text{ we get}\]
\[v + x\frac{dv}{dx} = v + \sin v\]
\[ \Rightarrow x\frac{dv}{dx} = v + \sin v - v\]
\[ \Rightarrow \frac{1}{\sin v}dv = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int \frac{1}{\sin v}dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \int cosec\ v\ dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \log \left| \tan \frac{v}{2} \right| = \log \left| x \right| + \log C\]
\[ \Rightarrow \log \left| \tan \frac{v}{2} \right| = \log \left| x \right| + \log C\]
\[ \Rightarrow \log \left| \tan \frac{v}{2} \right| = \log \left| Cx \right|\]
\[ \Rightarrow \tan \frac{v}{2} = Cx\]
\[\text{Putting }v = \frac{y}{x},\text{ we get}\]
\[ \Rightarrow \tan \left( \frac{y}{2x} \right) = Cx\]
\[\text{Hence, }\tan \left( \frac{y}{2x} \right) = Cx\text{ is the required solution.}\]
APPEARS IN
संबंधित प्रश्न
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Show that y = AeBx is a solution of the differential equation
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(ey + 1) cos x dx + ey sin x dy = 0
tan y dx + sec2 y tan x dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
Define a differential equation.
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Solve the following differential equation.
xdx + 2y dx = 0
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Solve:
(x + y) dy = a2 dx
Solve the differential equation xdx + 2ydy = 0
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
