Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Advertisements
उत्तर
y2 dx + (xy + x2 ) dy = 0
∴ (xy + x2 ) dy = - y2 dx
∴`dy/dx = (-y^2)/(xy+x^2)` ...(i)
Put y = tx ...(ii)
Differentiating w.r.t. x, we get
`dy/dx = t + x dt/dx` ...(iii)
Substituting (ii) and (iii) in (i), we get
`t + x dt/dx = (-t^2x^2)/(x.tx+x^2)`
∴`t + x dt/dx = (-t^2x^2)/(tx^2+x^2)`
∴ `t + x dt/dx = (-t^2x^2)/(x^2(t+1)`
∴ ` x dt/dx = (-t^2)/(t+1)-t`
∴ ` x dt/dx = (-t^2-t^2-t)/(t+1)`
∴ ` x dt/dx = (-(2t^2+t))/(t+1)`
∴ `(t+1)/(2t^2+t)dt = - 1/x dx`
Integrating on both sides, we get
`int (t+1)/(2t^2+t)dt = -int1/xdx`
∴`int (2t + 1 - t)/(t(2t+1)) dt = - int1/xdx`
∴`int1/tdt-int 1/ (2t+1) dt = -int1/ x dx`
∴ log | t | - `1/2` log |2t + 1| = - log |x| + log |c|
∴ 2log| t | - log |2t + 1| = - 2log |x| + 2 log |c|
∴ `2log |y/x | -log |(2y)/x+ 1 |= - 2log |x| + 2 log |c|`
∴ 2log |y| - 2log |x| - log |2y + x| + log |x|
= -2log |x| + 2log |c|
∴ log |y2 | + log |x| = log |c2 | + log |2y + x|
∴ log |y2 x| = log | c2 (x + 2y)|
∴ xy 2 = c2 (x + 2y)
APPEARS IN
संबंधित प्रश्न
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
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
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
(ey + 1) cos x dx + ey sin x dy = 0
y (1 + ex) dy = (y + 1) ex dx
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
y ex/y dx = (xex/y + y) dy
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Solve:
(x + y) dy = a2 dx
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
