Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Advertisements
उत्तर
(x2 − y2 ) dx + 2xy dy = 0
∴ 2xy dy = (y2 - x2) dx
∴ `dy/dx = (y^2 - x^2)/(2xy) ......(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^2)/(2tx^2)`
∴ `x dt/dx = (t^2 - 1)/(2t )- t = (-(1+t^2))/(2t)`
∴ `2t/(1+t^2) dt = - dx/x`
Integrating on both sides, we get
`int 2t/(1+t^2) dt = - int dx/x`
∴ log |1 + t2| = -log |x| + log |c|
∴`log | 1+y^2/x^2| = log |c/x|`
∴ `(x^2 + y^2)/x^2 = c/x`
∴ x2 + y2 = cx
APPEARS IN
संबंधित प्रश्न
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 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).
(x + y) (dx − dy) = dx + dy
3x2 dy = (3xy + y2) dx
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Define a differential equation.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Find the differential equation whose general solution is
x3 + y3 = 35ax.
The solution of `dy/dx + x^2/y^2 = 0` is ______
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.
y dx – x dy + log x dx = 0
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation
`x + y dy/dx` = x2 + y2
