Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`(x + y) dy/dx = 1`
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
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
C' (x) = 2 + 0.15 x ; C(0) = 100
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
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.
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Solve the differential equation:
dr = a r dθ − θ dr
Solve
`dy/dx + 2/ x y = x^2`
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
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
