Advertisements
Advertisements
प्रश्न
Solve the differential equation `"dy"/"dx" + 2xy` = y
Advertisements
उत्तर
Given equation is `"dy"/"dx" + 2xy` = y.
⇒ `"dy"/"dx"` = y – xy
⇒ `"dy"/"dx"` = y(1 –2x)
⇒ `"dy"/y` = (1 –2x)dx
Integrating both sides, we have
`int "dy"/"dx" = int (1 - 2x)"d"x`
⇒ log y = x – x2 + log c
⇒ log y – log c = x – x2
⇒ `log y/"c"` = x – x2
⇒ `y/"c" = "e"^(x - x^2)`
∴ y = `"c" . "e"^(x - x^2)`
Hence, the required solution is y = `"c" . "e"^(x - x^2)` .
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
x cos2 y dx = y cos2 x dy
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.
x2 dy + y (x + y) dx = 0
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
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 (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Solve the differential equation:
`e^(dy/dx) = x`
Solve:
(x + y) dy = a2 dx
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
