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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in `x^2e^x`.
Integrate the function in x cos-1 x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/sqrt(x^2 - 8x - 20) "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Which function is an example under priority \(I\) in the LIATE rule?
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
Repeated parts may be needed for which pair of integrals?
