Advertisements
Advertisements
Question
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Advertisements
Solution
`("d"y)/("d"x)` = x2y + y
= (x2 + 1)y
∴ `1/y "d"y` = (x2 + 1) dx
Integrating on both sides, we get
`int 1/y "d"y = int(x^2 + 1) "d"x`
∴ log |y| = `x^3/3 + x + c`
APPEARS IN
RELATED QUESTIONS
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
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 following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
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 \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
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?
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
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
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
The solution of `dy/dx + x^2/y^2 = 0` is ______
y2 dx + (xy + x2)dy = 0
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
Solve the differential equation
`y (dy)/(dx) + x` = 0
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
