Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
`e^x tan y dx = (e^x - 2)sec^2y dy`
Using variable seprable
`((sec^2y)/(tan y)) dy = (e^x/(e^x - 2))dx`
Integrating both side
`int ((sec^2 y)/(tany)) dy = int (e^x/(e^x - 2)) dx`
Let tan y=p ⇒ sec2y dy = dp and (ex−2) = q ⇒ exdx =dq
`int (dp)/p = int (dq)/q`
In (p) = In (q) + In(c), where c is constant of integration
p = qc
Replacing the values
`tan y = c(x^x - 2)`
When x = 0, `y = pi/4`
1 = c(-1)
c = -1
`tan y = 2 - e^x`
`y = tan^(-1) (2 - e^x)`
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:
(ex + e–x) dy – (ex – e–x) dx = 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:
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, -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.
In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
The general solution of the differential equation `dy/dx = e^(x+y)` is ______.
Find the particular solution of the differential equation:
`y(1+logx) dx/dy - xlogx = 0`
when y = e2 and x = e
Solve the equation for x:
sin-1x + sin-1(1 - x) = cos-1x, x ≠ 0
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
Solve
`y log y dy/dx + x – log y = 0`
State whether the following statement is True or False:
A differential equation in which the dependent variable, say y, depends only on one independent variable, say x, is called as ordinary differential equation
Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.
Solve the differential equation `"dy"/"dx" + 1` = ex + y.
Solve: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]
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 ______.
