Advertisements
Advertisements
Question
Solution
In this differential equation, the order of the highest order derivative is 2 and its power is 1. So, it is a differential equation of order 2 and degree 1.
It is a linear differential equation.
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
(y + xy) dx + (x − xy2) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
3x2 dy = (3xy + y2) dx
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
A population grows at the rate of 5% per year. How long does it take for the population to double?
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
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.
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
In the following example, verify that the given function is a solution of the corresponding differential equation.
Solution | D.E. |
xy = log y + k | y' (1 - xy) = y2 |
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.
`xy dy/dx = x^2 + 2y^2`
The solution of `dy/ dx` = 1 is ______
State whether the following is True or False:
The integrating factor of the differential equation `dy/dx - y = x` is e-x
Solve the differential equation:
`e^(dy/dx) = x`
Solve:
(x + y) dy = a2 dx
y dx – x dy + log x dx = 0
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
The function y = ex is solution ______ of differential equation
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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`
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
Solve the differential equation
`x + y dy/dx` = x2 + y2