Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Advertisements
उत्तर
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
∴ x2 (1 - y) dy = - y2 (1 + x) dx
∴ `((1-y)/y^2)dy = - ((1+x)/x^2)dx`
Integrating on both sides, we get
`int(1/y^2- 1/y) dy = - int (1/x^2+1/x)dx`
∴ `-1/y - log |y| = - (-1/x + log | x |)+c`
∴`(-1)/y - log |y| = 1/x - log | x |+c`
∴ `log | x | - log | y | = 1/x + 1/y + c`
APPEARS IN
संबंधित प्रश्न
Integrate the function in `e^x (1/x - 1/x^2)`.
Find :
`∫(log x)^2 dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : log (x2 + 1)
Evaluate the following.
`int x^2 e^4x`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int (sinx)/(1 + sin x) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
∫ log x · (log x + 2) dx = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
`int(logx)^2dx` equals ______.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`int1/(x+sqrt(x)) dx` = ______
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate `int tan^-1x dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
