Advertisements
Advertisements
प्रश्न
x2y dx – (x3 + y3) dy = 0
Advertisements
उत्तर
x2y dx – (x3 + y3) dy = 0
∴ x2y dx – (x3 + y3) = dy
∴ `dy/dx = (x^2y)/(x^3 + y^3)` …(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 . tx)/(x^3 + t^3 x^3)`
∴ `t + x dt/dx = (x^3.t)/(x^3(1+t^3))`
∴ `x dt/dx = t/(1+t^3) - t`
∴ `x dt/dx = (t-t-t^4)/(1+t^3)`
∴ `x dt/dx = (-t^4)/(1+t^3)`
∴ `(1+t^3)/t^4dt = - dx/x`
Integrating on both sides, we get
`int(1+t^3)/t^4 dt = - int 1/x dx `
∴ `int (1/t^4 + 1/t) dt = - int 1/x dx`
∴ `int t^-4 dt + int 1/t dt = - int1/x dx`
∴ `t^3/-3 + log |t| = - log |x| + c`
∴ `-1/(3t^3)+ log | t | = - log |x| + c`
∴ `- 1/3 . 1/(y/x)^3 + log|y/x| = - log |x| + c`
∴`x^3/(3y^3) + log |y| - log |x| = - log |x| + c`
∴`log |y| - x^3/ (3y^3) = c`
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
x cos y dy = (xex log x + ex) dx
(1 − x2) dy + xy dx = xy2 dx
y (1 + ex) dy = (y + 1) ex dx
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
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.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
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.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation y1 y3 = y22 is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve the differential equation:
`e^(dy/dx) = x`
Solve
`dy/dx + 2/ x y = x^2`
Solve: `("d"y)/("d"x) + 2/xy` = x2
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 `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
