Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Advertisements
उत्तर १
`"x" "dy"/"dx" - "y" + "x sin"("y"/"x") = 0` ...(1)
Put y = vx
∴ `"dy"/"dx" = "v + x" "dv"/"dx" and "y"/"x" = "v"`
∴ equation (1) becomes,
`x("v + x""dv"/"dx") - "vx + x sin v" = 0`
∴ `"vx" + "x"^2 "dv"/"dx" - "vx" + "x sin v" = 0`
∴ `"x"^2 "dv"/"dx" + "x sin v" = 0`
∴ `1/"sin v" "dv" + 1/"x" "dx" = 0`
Integrating, we get
∴ `int "cosec v dv" + int1/"x" "dx" = "c"_1`
∴ `log |"cosec v - cot v"| + log |"x"| = log "c"`, where c1 = log c
∴ `log |"x" ("cosec v" - "cot v")| = log "c"`
∴ `"x"(1/(sin"v") - (cos "v")/(sin"v")) = "c"`
∴ x(1 - cos v) = c sin v
∴ `"x"[1 - cos("y"/"x")] = "c sin"("y"/"x")`
This is the general solution.
उत्तर २
`x * dy/dx - y + x * sin(y/x) = 0`
Put y = vx
⇒ `dy/dx = v + x (dv)/dx`
∴ The given equation becomes,
`x (v + x (dv)/dx) - vx + x sin v = 0`
∴ `vx + x^2 (dv)/dx - vx + x sin v = 0`
∴ `x^2 (dv)/dx = -x sin v`
∴ `(dv)/(sin v) = - dx/x` ...(variable separable form)
Integrating on both sides, we get,
`int cosec v dv = -int 1/x dx`
∴ log (cosec v − cot v) = − log x + log c
∴ log (cosec v − cot v) = `log (c/x)`
∴ cosec v − cot v = `c/x`
∴ `x [cosec (y/x) - cot (y/x)] = c` is the required solution.
संबंधित प्रश्न
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 – y) dy – (x + y) dx = 0
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.
`y dx + x log(y/x)dy - 2x dy = 0`
Show that the given differential equation is homogeneous and solve them.
`(1+e^(x/y))dx + e^(x/y) (1 - x/y)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`
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
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:
(xy − y2) dx − x2 dy = 0, y(1) = 1
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.
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 differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
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 ______.
Read the following passage:
|
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
The solution of the equation `dy/dx = (3x − 4y − 2)/(3x − 4y − 3)` is ______.
For the substitution \[x=vy\], which differentiated form is correct?
Which dependence is checked on the right-hand side of a homogeneous differential equation?
Which substitutions are chosen for a homogeneous differential equation?
What form should the equation be reduced to before integration?
What replacement obtains the final answer after using a homogeneous substitution?
For \[F(x,y)=\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\], what is \[F(\lambda x,\lambda y)\]?
From \[v+x\frac{dv}{dx}=\frac{v\cos v+1}{\cos v}\], what is \[x\frac{dv}{dx}\]?
Which final answer results after replacing \[v=\frac{y}{x}\] in \[\sin v=\ln|Cx|\]?
