Advertisements
Advertisements
प्रश्न
Solve
`y log y dy/dx + x – log y = 0`
Advertisements
उत्तर
`y log y dy/dx + x – log y = 0`
∴ `dx/dy + 1/(y log y) x= 1/y`
The given equation is of the form `dx/dy + px = Q`
where, `P = 1 /(y logy )and Q = 1/y`
∴ `I.F. = e ^(int^pdy) = e^(int^(1/(ylogy)) dy) = e ^(log | log y|) = log y`
∴ Solution of the given equation is
`x(I.F.) =int Q (I.F.) dy + c_1`
∴ `x.log y = int 1/y log y dy + c_1`
In R. H. S., put log y = t
Differentiating w.r.t. x, we get
`1/y dy = dt`
∴ `x log y =int t dt + c_1 = t^2/2 + c_1`
∴`x log y =(logy)^2/2 + c_1`
∴ 2x log y = (log y)2 + c …[2c1 = c]
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`dy/dx + y = 1(y != 1)`
For the differential equation, find the general solution:
sec2 x tan y dx + sec2 y tan x dy = 0
For the differential equation, find the general solution:
`dy/dx = (1+x^2)(1+y^2)`
For the differential equation, find the general solution:
y log y dx - x dy = 0
For the differential equation, find the general solution:
`x^5 dy/dx = - y^5`
For the differential equation, find the general solution:
`dy/dx = sin^(-1) x`
For the differential equation find a particular solution satisfying the given condition:
`x(x^2 - 1) dy/dx = 1` , y = 0 when x = 2
For the differential equation find a particular solution satisfying the given condition:
`cos (dx/dy) = a(a in R); y = 1` when x = 0
For the differential equation find a particular solution satisfying the given condition:
`dy/dx` = y tan x; y = 1 when x = 0
Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = e x sin x.
Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point.
The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 doubles itself in 10 years (loge 2 = 0.6931).
In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
The general solution of the differential equation `dy/dx = e^(x+y)` is ______.
Find the particular solution of the differential equation ex tan y dx + (2 – ex) sec2 y dy = 0, give that `y = pi/4` when x = 0
Find the particular solution of the differential equation `dy/dx + 2y tan x = sin x` given that y = 0 when x = `pi/3`
Fill in the blank:
The integrating factor of the differential equation `dy/dx – y = x` is __________
Verify y = log x + c is a solution of the differential equation
`x(d^2y)/dx^2 + dy/dx = 0`
Solve
y dx – x dy = −log x dx
The resale value of a machine decreases over a 10 year period at a rate that depends on the age of the machine. When the machine is x years old, the rate at which its value is changing is ₹ 2200 (x − 10) per year. Express the value of the machine as a function of its age and initial value. If the machine was originally worth ₹1,20,000, how much will it be worth when it is 10 years old?
Solve
`y log y dx/ dy = log y – x`
Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.
Solve the differential equation `"dy"/"dx" + 1` = ex + y.
Find the equation of the curve passing through the (0, –2) given that at any point (x, y) on the curve the product of the slope of its tangent and y-co-ordinate of the point is equal to the x-co-ordinate of the point.
Solve the following differential equation
x2y dx – (x3 + y3)dy = 0
