Advertisements
Advertisements
प्रश्न
`xy dy/dx = x^2 + 2y^2`
Advertisements
उत्तर
`xy dy/dx = x^2 + 2y^2`
∴ `dy/dx = x^2 + (2y^2)/ (xy)` …(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 = (x^2 + 2t^2 x^2)/ (x(tx))`
∴ `t + x dt/dx = (x^2 (1+2t^2))/(x^2t)`
∴ `x dt/dx (1+2t^2)/t - t = (1+ t^2)/t`
∴ `t/(1+t^2) dt = 1/x dx`
Integrating on both sides, we get
`1/2 int (2t)/(1+t^2) dt = int dx/x`
∴ `1/2 log|1 + t^2| = log |x| + log |c|`
∴ log |1 + t2 | = 2 log |x| + 2log |c|
= log |x2 | + log |c2|
∴ log |1 + t2 | = log |c2 x2|
∴ 1 + t2 = c2x2
∴ `1 + y^2/x^2 = c^2x^2`
∴ x2 + y2 = c2 x4
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Verify that y = cx + 2c2 is a solution of the differential equation
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
y2 dx + (x2 − xy + y2) dy = 0
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the differential equation:
`e^(dy/dx) = x`
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Solve the following differential equation y2dx + (xy + x2) dy = 0
Solve: ydx – xdy = x2ydx.
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
Solve the differential equation
`y (dy)/(dx) + x` = 0
