Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Advertisements
उत्तर
`(dθ)/dt = − k (θ − θ_0)`, k is constant.
∴`(dθ)/ (θ − θ_0) = -k dt`
Integrating on both sides, we get
`int(dθ)/ (θ − θ_0) = -k int dt`
∴ log |θ − θ0| = - kt + c
∴ θ − θ0 = `e ^(-kt+c)`
APPEARS IN
संबंधित प्रश्न
Show that y = AeBx is a solution of the differential equation
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 y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
The solution of `dy/ dx` = 1 is ______.
The solution of `dy/dx + x^2/y^2 = 0` is ______
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
