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
Show that the given differential equation is homogeneous and solve them.
(x2 – y2) dx + 2xy dy = 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.
`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`
For the differential equation find a particular solution satisfying the given condition:
x2 dy + (xy + y2) dx = 0; y = 1 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.
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:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
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
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 differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
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`
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
If \[F(\lambda x,\lambda y)=F(x,y)\] for any non-zero constant \[\lambda\], what is the degree of \[F(x,y)\]?
For the substitution \[y=vx\], which differentiated form is correct?
After using \[y=vx\] and writing the right-hand side as \[g(v)\], which separable form is obtained?
Which substitutions are chosen for a homogeneous differential equation?
What form should the equation be reduced to before integration?
For \[F(x,y)=\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\], what is \[F(\lambda x,\lambda y)\]?
Which final answer results after replacing \[v=\frac{y}{x}\] in \[\sin v=\ln|Cx|\]?
Which sequence correctly states the essential actions for obtaining the final answer of a homogeneous differential equation?
