Advertisements
Advertisements
प्रश्न
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Advertisements
उत्तर
Given equation is `"dy"/"dx"` = e–2y
⇒ `"dy"/"e"^(-2y)` = dx
⇒ `"e"^(2y) * "d"y` = dx
Integrating both sides, we get
`int "e"^(2y) "d"y = int "d"x`
⇒ `1/2 "e"^(2y)` = x + c
Put y = 0 and x = 5
⇒ `1/2 "e"^0` = 5 + c
⇒ c = `1/2 - 5 = - 9/2`
Now putting y = 3, we get
`1/2 "e"^6 = x - 9/2`
⇒ x = `1/2 "e"^6 + 9/2`
Hence the required value of x =`1/2 ("e"^6 + 9)`.
APPEARS IN
संबंधित प्रश्न
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
C' (x) = 2 + 0.15 x ; C(0) = 100
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
y ex/y dx = (xex/y + y) dy
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
The solution of `dy/ dx` = 1 is ______
Solve
`dy/dx + 2/ x y = x^2`
y dx – x dy + log x dx = 0
Solve: `("d"y)/("d"x) + 2/xy` = x2
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation `"dy"/"dx" + 2xy` = y
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
