Advertisements
Advertisements
Question
For the differential equation, find the general solution:
`x log x dy/dx + y= 2/x log x`
Solution
The given equation
`x log x dy/dx + y = 2/x log x`
or `dy/dx + y/(x log x) = 1/x^2` ....(i)
Comparing with `dy/dx + Py = Q`,
`P = 1/(x log x)` and `Q = 2/x^2`
∴ `I.F. = e^(intP dx) = e^(int 1/(x log x)dx)`
`= e^(log(log x)) = log x`
Hence the required solution
∴ `y × I.F. = int I.F. xx Q dx + C`
`=> y log x = 2 int 1/x^2 (log x) dx + C`
`=> y log x = 2 [log x (- 1/x) - int 1/x ((- 1)/x) dx] + C`
`=> y log x = (- 2)/x log x + 2 int 1/x^2 dx + C`
`=> y log x = (- 2)/x log x - 2/x + C`
⇒ `y = log x = (- 2)/x (1 + log |x|) + C`
APPEARS IN
RELATED QUESTIONS
For the differential equation, find the general solution:
`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`
For the differential equation, find the general solution:
`x dy/dx + 2y= x^2 log x`
For the differential equation given, find a particular solution satisfying the given condition:
`dy/dx - 3ycotx = sin 2x; y = 2` when `x = pi/2`
Solve the differential equation `(tan^(-1) x- y) dx = (1 + x^2) dy`
Solve the differential equation `x dy/dx + y = x cos x + sin x`, given that y = 1 when `x = pi/2`
Find the general solution of the differential equation \[x\frac{dy}{dx} + 2y = x^2\]
Solve the following differential equation:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]
Find the integerating factor of the differential equation `x(dy)/(dx) - 2y = 2x^2`
If f(x) = x + 1, find `"d"/"dx"("fof") ("x")`
Solve the following differential equation:
`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`
Solve the following differential equation:
`("x" + 2"y"^3) "dy"/"dx" = "y"`
Solve the following differential equation dr + (2r cot θ + sin 2θ) dθ = 0.
Solve the following differential equation:
y dx + (x - y2) dy = 0
Solve the following differential equation:
`(1 - "x"^2) "dy"/"dx" + "2xy" = "x"(1 - "x"^2)^(1/2)`
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.
Find the equation of the curve passing through the point `(3/sqrt2, sqrt2)` having a slope of the tangent to the curve at any point (x, y) is -`"4x"/"9y"`.
The integrating factor of `(dy)/(dx) + y` = e–x is ______.
Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 is ______.
The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.
The equation x2 + yx2 + x + y = 0 represents
Let y = y(x), x > 1, be the solution of the differential equation `(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`, with y(2) = `(1 + e^4)/(2e^4)`. If y(3) = `(e^α + 1)/(βe^α)`, then the value of α + β is equal to ______.
If y = y(x) is the solution of the differential equation, `(dy)/(dx) + 2ytanx = sinx, y(π/3)` = 0, then the maximum value of the function y (x) over R is equal to ______.
Let y = f(x) be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If f(x) satisfies xf'(x) = x2 + f(x) – 2, then the area bounded by f(x) with x-axis between ordinates x = 0 and x = 3 is equal to ______.
The solution of the differential equation `dx/dt = (xlogx)/t` is ______.
Solve the differential equation `dy/dx+2xy=x` by completing the following activity.
Solution: `dy/dx+2xy=x` ...(1)
This is the linear differential equation of the form `dy/dx +Py =Q,"where"`
`P=square` and Q = x
∴ `I.F. = e^(intPdx)=square`
The solution of (1) is given by
`y.(I.F.)=intQ(I.F.)dx+c=intsquare dx+c`
∴ `ye^(x^2) = square`
This is the general solution.
If sec x + tan x is the integrating factor of `dy/dx + Py` = Q, then value of P is ______.
The slope of tangent at any point on the curve is 3. lf the curve passes through (1, 1), then the equation of curve is ______.
Solve:
`xsinx dy/dx + (xcosx + sinx)y` = sin x
