Advertisements
Advertisements
Question
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
Advertisements
Solution
Given that: `x ("d"y)/("d"x) = y(log y – log x + 1)`
⇒ `x ("d"y)/("d"x) = y[log(y/x) + 1]`
⇒ `("d"y)/("d"x) = y/x[log(y/x) + 1]`
Since, it is a homogeneous differential equation.
∴ Put y = vx
⇒ `("d"y)/("d"x) = "v" + x * "dv"/"dx"`
∴ `"v" + x * "dv"/"dx" = "vx"/x[log("vx"/x) + 1]`
⇒ `"v" + x * "dv"/"dx" = "v"[log "v" + 1]`
⇒ `x * "dv"/"dx" = "v"[log "v" + 1] - "v"`
⇒ `x * "dv"/"dx"` = v ....[log v + 1 – 1]
⇒ `x * "dv"/"dx" = "v" * log "v"`
⇒ `"dv"/("v"log"v") = "dx"/x`
Integrating both sides, we get
`int "dv"/("v"log"v") = int "dx"/x`
Put log v = t on L.H.S.
`1/"v" "dv"` = dt
∴ `int "dt"/"t" = int "dx"/x`
`log|"t"| = log|x| + log"c"`
⇒ `log|log "v"| = log x"c"`
⇒ log v = xc
⇒ `log(y/x)` = xc
Hence, the required solution is `log(y/x)` = xc.
APPEARS IN
RELATED QUESTIONS
Solve the differential equation (x2 + y2)dx- 2xydy = 0
Find the particular solution of the differential equation:
2y ex/y dx + (y - 2x ex/y) dy = 0 given that x = 0 when y = 1.
Show that the given differential equation is homogeneous and solve them.
(x2 + xy) dy = (x2 + y2) dx
Show that the given differential equation is homogeneous and solve them.
`y' = (x + y)/x`
Show that the given differential equation is homogeneous and solve them.
(x – y) dy – (x + y) dx = 0
Show that the given differential equation is homogeneous and solve them.
`x^2 dy/dx = x^2 - 2y^2 + xy`
Show that the given differential equation is homogeneous and solve them.
`x dy/dx - y + x sin (y/x) = 0`
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x dy = 0`
For the differential equation find a particular solution satisfying the given condition:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
For the differential equation find a particular solution satisfying the given condition:
`2xy + y^2 - 2x^2 dy/dx = 0; y = 2` when x = 1
Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.
Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy, where C is parameter
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
Solve the following initial value problem:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
Solve the following initial value problem:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]
Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
Which of the following is a homogeneous differential equation?
Solve the following differential equation : \[\left[ y - x \cos\left( \frac{y}{x} \right) \right]dy + \left[ y \cos\left( \frac{y}{x} \right) - 2x \sin\left( \frac{y}{x} \right) \right]dx = 0\] .
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`x^2 dy/dx = x^2 + xy + y^2`
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
