Advertisements
Advertisements
प्रश्न
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
Advertisements
उत्तर
We have
\[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0 . . . . . \left( 1 \right)\]
\[Now, \]
\[y = e^{- 3x} \]
\[ \Rightarrow \frac{dy}{dx} = - 3 e^{- 3x} \]
\[\Rightarrow\frac{d^2y}{dx^2}=9e^{-3x}\]
\[\text{Putting the values of }\frac{d^2 y}{d x^2}, \frac{dy}{dx}\text{ and y in (1), we get}\]
\[LHS = 9 e^{- 3x} - 3 e^{- 3x} - 6 e^{- 3x} \]
\[ = 0\]
\[ = RHS\]
Thus, y = e−3x is the solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
(1 + x2) dy = xy dx
x cos2 y dx = y cos2 x dy
(1 − x2) dy + xy dx = xy2 dx
tan y dx + sec2 y tan x dy = 0
(x + y) (dx − dy) = dx + dy
3x2 dy = (3xy + y2) dx
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
The function y = ex is solution ______ of differential equation
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
