Advertisements
Advertisements
Question
Solve the differential equation `"dy"/"dx" + 2xy` = y
Advertisements
Solution
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
RELATED QUESTIONS
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
(1 + x2) dy = xy dx
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
(x2 − y2) dx − 2xy dy = 0
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 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?
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.
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
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).
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
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?
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
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`
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
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
