Advertisements
Advertisements
Question
Solve the following differential equation.
`(x + y) dy/dx = 1`
Advertisements
Solution
`(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
RELATED QUESTIONS
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
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
xy dy = (y − 1) (x + 1) dx
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
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.
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
(x + y) (dx − dy) = dx + dy
3x2 dy = (3xy + y2) dx
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?
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
y2 dx + (x2 − xy + y2) dy = 0
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
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 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` |
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
