Advertisements
Advertisements
प्रश्न
Find the solution of `"dy"/"dx"` = 2y–x.
Advertisements
उत्तर
The given differential equation is
`"dy"/"dx"` = 2y–x
⇒ `"dy"/"dx" = 2^y/2^x`
Separating the variables, we get
`"dy"/2^y = "dx"/2^x`
⇒ `2^-y "d"y = 2^-x "d"x`
Integrating both sides, we get
`int 2^-y "d"y = int 2^-x "d"x`
`(-2^-y)/log2 = (-2^-x)/log2 + "c"`
⇒ `-2^-y = -2^-x + "c" log 2`
⇒ `-2^-y + 2^-x = "c" log 2`
⇒ `2^-x - 2^-y` = k .....[Where c log 2 = k]
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`dy/dx = (1 - cos x)/(1+cos x)`
For the differential equation, find the general solution:
`dy/dx = sqrt(4-y^2) (-2 < y < 2)`
For the differential equation, find the general solution:
(ex + e–x) dy – (ex – e–x) dx = 0
For the differential equation, find the general solution:
y log y dx - x dy = 0
For the differential equation, find the general solution:
`dy/dx = sin^(-1) x`
For the differential equation, find the general solution:
ex tan y dx + (1 – ex) sec2 y dy = 0
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.
For the differential equation `xy(dy)/(dx) = (x + 2)(y + 2)` find the solution curve passing through the point (1, –1).
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 equation of the curve passing through the point `(0,pi/4)`, whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Solve the equation for x:
sin-1x + sin-1(1 - x) = cos-1x, x ≠ 0
Solve the differential equation `"dy"/"dx" = 1 + "x"^2 + "y"^2 +"x"^2"y"^2`, given that y = 1 when x = 0.
Verify y = log x + c is a solution of the differential equation
`x(d^2y)/dx^2 + dy/dx = 0`
Solve the differential equation:
`dy/dx = 1 +x+ y + xy`
Solve `dy/dx = (x+y+1)/(x+y-1) when x = 2/3 and y = 1/3`
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`
Find the differential equation of all non-vertical lines in a plane.
Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.
Solve the differential equation `"dy"/"dx" + 1` = ex + y.
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 method of solving a differential equation can be used to solve `"dy"/"dx" = "k"(50 - "y")`?
Solve the following differential equation
x2y dx – (x3 + y3)dy = 0
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4th day is 30, then number of infected studetns on 8th day will be ______.
