Advertisements
Advertisements
प्रश्न
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^2 x^2)/(x.tx + x^2)`
∴ `t + x dt/dx = (-t^2 x^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 = - int 1/x dx`
∴ `int (2t +1 - t)/(t(2t+1)) dt = - int 1/x dx`
∴ `int 1/t dt - int 1/(2t + 1) dt = -int 1/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 |y2x| = log |c2(x + 2y)|
∴ xy2 = c2 (x + 2y)
APPEARS IN
संबंधित प्रश्न
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
The solution of `dy/dx + x^2/y^2 = 0` is ______
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
