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 = \[\frac{a}{x} + b\] 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 |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
(ey + 1) cos x dx + ey sin x dy = 0
xy dy = (y − 1) (x + 1) dx
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
y2 dx + (xy + x2)dy = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
In mathematics and science, what primary purpose do differential equations serve?
