Advertisements
Advertisements
प्रश्न
Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`
Advertisements
उत्तर
Consider the given differential equation:
`x logx dy/dx+y=2log x`
Dividing the above equation by xlogx, we have,
`(x logx)/(x logx)dy/dx+y/(x logx)=(2log x)/(x logx)`
`=>dy/dx+y/(x logx)=1/x ........(1)`
Consider the general linear differential equation
`dy/dx+Py=Q,` where P and Q are functions of x.
Comparing equation (1) and the general equation, we have,
`P(x)=1/xlogx and Q(x)=2/x`
The integrating factor is given by the formula `e^(intPdx)`
Thus `I.F=e^(intPdx)=e^(intdx/(xlogx))`
Consider `I=int dx/(xlogx)`
Substituting logx=t; dx/x=dt
Thus `I=intdt/t=log(t)=log(logx)`
Hence ` I.F=e^(intdx/(xlogx))=e^(log(logx))=logx`
APPEARS IN
संबंधित प्रश्न
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
\[\frac{dy}{dx}\] + y cot x = x2 cot x + 2x
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
Solve the differential equation: (x + 1) dy – 2xy dx = 0
Solve the following differential equation :
`"dy"/"dx" + "y" = cos"x" - sin"x"`
`("e"^(-2sqrt(x))/sqrt(x) - y/sqrt(x))("d"x)/("d"y) = 1(x ≠ 0)` when written in the form `"dy"/"dx" + "P"y` = Q, then P = ______.
`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.
Integrating factor of the differential equation of the form `("d"x)/("d"y) + "P"_1x = "Q"_1` is given by `"e"^(int P_1dy)`.
Correct substitution for the solution of the differential equation of the type `("d"y)/("d"x) = "f"(x, y)`, where f(x, y) is a homogeneous function of zero degree is y = vx.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
Which of the following solutions may be used to find the number of children who have been given the polio drops?
If α, β are different values of x satisfying the equation a cos x + b sinα x = c, where a, b and c are constants, then `tan ((alpha + beta)/2)` is
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`
Let y = y(x) be the solution of the differential equation `(dy)/(dx) + (sqrt(2)y)/(2cos^4x - cos2x) = xe^(tan^-1(sqrt(2)cost2x)), 0 < x < π/2` with `y(π/4) = π^2/32`. If `y(π/3) = π^2/18e^(-tan^-1(α))`, then the value of 3α2 is equal to ______.
If y = y(x) is the solution of the differential equation `(1 + e^(2x))(dy)/(dx) + 2(1 + y^2)e^x` = 0 and y(0) = 0, then `6(y^'(0) + (y(log_esqrt(3))))^2` is equal to ______.
Let y = y(x) be the solution of the differential equation `xtan(y/x)dy = (ytan(y/x) - x)dx, -1 ≤ x ≤ 1, y(1/2) = π/6`. Then the area of the region bounded by the curves x = 0, x = `1/sqrt(2)` and y = y(x) in the upper half plane is ______.
Let y = y(x) be the solution of the differential equation, `(x^2 + 1)^2 ("dy")/("d"x) + 2x(x^2 + 1)"y"` = 1, such that y(0) = 0. If `sqrt("ay")(1) = π/32` then the value of 'a' is ______.
