Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`(x + y) dy/dx = 1`
Advertisements
उत्तर
`(x + y) dy/dx = 1`
∴ `dy/dx = 1/(x + y)`
∴ `dx/dy = x + y`
∴ `dx/dy − x = y`
∴ `dx/dy + (− 1)x = y` ...(I)
The given equation is of the form `dx/dy + Px = Q`
where, P = − 1 and Q = y
∴ `"I.F." = e int ^"pdy" = e int ^("(−1)dy") = e^-"y"`
∴ Solution of the given equation is
`"x (I.F.)" = int "Q (I.F.) dy" + C`
∴ `"xe"^(−"y") = ubrace(int y.e^(−y))_(("I")) dy + C` ...(II)
Let I = `int y. e^(−y) dy`
Using Integration by parts,
I = `y int e^(−y) dy - int [ d/dy y int e^(−y) dy] dy`
I = `y (e^(−y))/(-1) − int 1. (e^(−y))/(-1) dy`
I = `− y. e^(−y) + int e^(−y) dy`
I = `− y. e^(−y) + e^(−y)/(- 1) dy`
I = `− y. e^(−y) − e^(−y)`
Putting value of I in (2),
∴ `"xe"^(−"y") = int y.e^(−y) dy + C`
∴ `"xe"^(−"y") = − y. e^(−y) − e^(−y) + C`
Dividing by e−y,
∴ x = − y − 1 + Cey
∴ x + y + 1 = Cey
APPEARS IN
संबंधित प्रश्न
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
C' (x) = 2 + 0.15 x ; C(0) = 100
Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, 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.
(x + y) (dx − dy) = dx + dy
y ex/y dx = (xex/y + y) dy
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
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.
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
The solution of the differential equation y1 y3 = y22 is
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
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` |
y dx – x dy + log x dx = 0
Solve the differential equation `("d"y)/("d"x) + y` = e−x
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Solve: ydx – xdy = x2ydx.
