Advertisements
Advertisements
Question
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Advertisements
Solution 1
`"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.
Solution 2
`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.
RELATED QUESTIONS
Show that the differential equation 2yx/y dx + (y − 2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.
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.
(x2 – y2) dx + 2xy dy = 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:
`dy/dx - y/x + cosec (y/x) = 0; y = 0` 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
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
Which of the following is a homogeneous differential equation?
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(x2 + 3xy + y2) dx − x2 dy = 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:
(xy − y2) dx − x2 dy = 0, y(1) = 1
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) 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.
Solve the following differential equation:
`(1 + 2"e"^("x"/"y")) + 2"e"^("x"/"y")(1 - "x"/"y") "dy"/"dx" = 0`
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
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
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
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 ______.
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?
For the substitution \[x=vy\], which differentiated form is correct?
Which dependence is checked on the right-hand side of a homogeneous differential equation?
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 \[\sin v=\ln|Cx|\]?
Which sequence correctly states the essential actions for obtaining the final answer of a homogeneous differential equation?
