Advertisements
Advertisements
प्रश्न
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Advertisements
उत्तर
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y(2) = \pi\]
It is a homogeneous equation . put y = vx
\[\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[\text{ so,} v + x\frac{dv}{dx} = \frac{vx}{x} - \sin\left( \frac{vx}{x} \right)\]
\[x\frac{dv}{dx} = - \sin v\]
\[\frac{dv}{\sin v} = - \frac{dx}{x}\]
\[\text{ cosec }(v)dv = - \frac{dx}{x}\]
Integraing both sides we get,
\[\log(\text{cosec }(v) - cot(v)) = - \log x + \log c\]
\[log\left( \text{cosec }\left( \frac{y}{x} \right) - cot\left( \frac{y}{x} \right) \right) = - log x + log c\]
\[\text{Putting the values }x = 2\text{ and }y = \pi \]
\[log\left(\text{cosec }\left( \frac{\pi}{2} \right) - cot\left( \frac{\pi}{2} \right) \right) = - log 2 + log c\]
\[c = 0\]
\[log\left( \text{cosec }\left( \frac{y}{x} \right) - cot\left( \frac{y}{x} \right) \right) = - log x\]
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.
(x2 + xy) dy = (x2 + y2) dx
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.
`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`
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
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(x2 + 3xy + y2) dx − x2 dy = 0
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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 differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2 y/x`.
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
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 ______.
The solution of the differential equation y2 dx + (x2 − xy + y2)dy = 0 is ______.
A function \[F(x,y)\] is homogeneous of degree \[n\] when which condition holds?
For the substitution \[y=vx\], which differentiated form is correct?
Which dependence is checked on the right-hand side of a homogeneous differential equation?
What form should the equation be reduced to before integration?
Writing \[x\cos\left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos\left(\frac{y}{x}\right)+x\] in standard form gives which expression?
From \[v+x\frac{dv}{dx}=\frac{v\cos v+1}{\cos v}\], what is \[x\frac{dv}{dx}\]?
Which sequence correctly states the essential actions for obtaining the final answer of a homogeneous differential equation?
