Advertisements
Advertisements
Question
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) = x + y`
Advertisements
Solution
`x ("d"y)/("d"x) = x + y`
⇒ `("d"y)/("d"x) = (x + y)/x` .......(1)
It is a homogeneous differential equation, Same degree in x and y
Put y = vx and `("d"y)/("d"x) = "v" + x ("d"y)/("d"x)`
Equation (1)
⇒ `"v" + x ("d"y)/("d"x) = (x + "v"x)/x = (x(1 + "v"))/x`
`"v" + x "dv"/("d"x) = (1 + "v")`
⇒ `x ("d"y)/("d"x) = 1 + "v" - "v"`
`x "dv"/("d"x)` = 1
dv = `1/x "d"x`
Integrating on both sides
`int 1/x "d"x = int "dv"`
log x = v + c
⇒ x = `"e"^("v" + "c")`
x = ev. ec
x = ev. c
⇒ x = cev ......[⇒ `"v" = y/x]`
⇒ x = `"ce"^(y/x)`
APPEARS IN
RELATED QUESTIONS
Solve the following differential equation:
`y"d"x + (1 + x^2)tan^-1x "d"y`= 0
Solve the following differential equation:
`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`
Solve the following differential equation:
`(x^3 + y^3)"d"y - x^2 y"d"x` = 0
Choose the correct alternative:
The number of arbitrary constants in the general solutions of order n and n +1are respectively
Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0
Find the curve whose gradient at any point P(x, y) on it is `(x - "a")/(y - "b")` and which passes through the origin
Solve the following homogeneous differential equation:
`(x - y) ("d"y)/("d"x) = x + 3y`
Solve the following homogeneous differential equation:
The slope of the tangent to a curve at any point (x, y) on it is given by (y3 – 2yx2) dx + (2xy2 – x3) dy = 0 and the curve passes through (1, 2). Find the equation of the curve
Choose the correct alternative:
The integrating factor of the differential equation `("d"y)/("d"x) + "P"x` = Q is
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
