Advertisements
Advertisements
प्रश्न
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
Advertisements
उत्तर
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
संबंधित प्रश्न
Solve the differential equation (x2 + y2)dx- 2xydy = 0
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
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^2 dy/dx = x^2 - 2y^2 + xy`
Show that the given differential equation is homogeneous and solve them.
`x dy - y dx = sqrt(x^2 + y^2) dx`
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`
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:
`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "when" x = 1`
For the differential equation find a particular solution satisfying the given condition:
`dy/dx - y/x + cosec (y/x) = 0; y = 0` 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.
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 0
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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.
Which of the following is a homogeneous differential equation?
Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
Let the solution curve of the differential equation `x (dy)/(dx) - y = sqrt(y^2 + 16x^2)`, y(1) = 3 be y = y(x). Then y(2) is equal to ______.
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
