Advertisements
Advertisements
Question
Find the solution of `"dy"/"dx"` = 2y–x.
Solution
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
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:
sec2 x tan y dx + sec2 y tan 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 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.
At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (- 4, -3). Find the equation of the curve given that it passes through (-2, 1).
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).
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.
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 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`
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: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]
The solution of the differential equation, `(dy)/(dx)` = (x – y)2, when y (1) = 1, is ______.
