Advertisements
Advertisements
Question
Solve the differential equation:
`e^(dy/dx) = x`
Advertisements
Solution
`e^(dy/dx) = x`
∴ `dy/dx = log x`
∴ dy = log x dx
Integrating on both sides, we get
`int dy = int (logx)1 dx`
∴ `y = log x int1dx - int [ d/dx(logx)int1dx] dx`
= `x log x -int 1/x. x dx`
= `x log x -int dx`
∴ y = x log x - x + c
APPEARS IN
RELATED QUESTIONS
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
(sin x + cos x) dy + (cos x − sin x) dx = 0
(y + xy) dx + (x − xy2) dy = 0
x2 dy + y (x + y) dx = 0
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
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.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c1
∴ e–2y = – 2sin x – 2c1
∴ `square` = c, where c = – 2c1
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
