Advertisements
Advertisements
Question
For the differential equation, find the general solution:
`dy/dx + y = 1(y != 1)`
Solution
`dy/dx + y = 1`
⇒ `dy/dx = - (y - 1)`
⇒ `dy/(y - 1) = - dx` ....(1)
Integrating (1) both sides, we get
⇒ `intdy/(y - 1) = - intdx`
⇒ log (y - 1) = -x + C1
⇒ y - 1 = e-x+C1
⇒ y = 1 + e-x . eC1
Hence, y = 1 + Ce-x, which is the required solution
APPEARS IN
RELATED QUESTIONS
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:
`x^5 dy/dx = - y^5`
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:
`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).
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 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).
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?
Find the particular solution of the differential equation:
`y(1+logx) dx/dy - xlogx = 0`
when y = e2 and x = e
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`
Solve the equation for x:
sin-1x + sin-1(1 - x) = cos-1x, x ≠ 0
Verify y = log x + c is a solution of the differential equation
`x(d^2y)/dx^2 + dy/dx = 0`
Solve `dy/dx = (x+y+1)/(x+y-1) when x = 2/3 and y = 1/3`
Solve
`y log y dy/dx + x – log y = 0`
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`
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
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")`?
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
