Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
The given equation
`2xy + y^2 - 2x^2 dy/dx = 0`
or `dy/dx = (2xy + y^2)/(2x^2)`
`= y/x + 1/2(y/x)^2` ....(i)
Clearly, this equation is a differential equation.
∴ putting y = vx
`dy/dx = v + x (dv)/dx` ....(in equation (i))
`v + x (dv)/dx = v + 1/2 v^2`
`=> x (dv)/dx = 1/2 v^2`
`=> 2 xx 1/v^2 (dv) = 1/x dv`
On integrating,
`2 int 1/v^2 dv = int 1/x dv - 2/v = log |x| + C`
So, on substituting `(y/x)` in place of v,
`- (2x)/y = log |x| + C` ....(ii)
Given y = 2 if x = 1 ...[from equation (ii)]
`(- 2)/2 = log1 + C`
- 1 = 0 + C
⇒ C = - 1
On putting C = – 1 in equation (ii)
`(- 2x)/y = log |x| - 1`
y = `(2x)/(1 - log |x|)`
APPEARS IN
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.
`x dy - y dx = sqrt(x^2 + y^2) dx`
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:
`[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
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
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.
(x2 + 3xy + y2) dx − x2 dy = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 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
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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.
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
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:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
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`.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
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
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy = (2xy + y2)dx, then `f(1/2)` is equal to ______.
The solution of the differential equation y2 dx + (x2 − xy + y2)dy = 0 is ______.
