Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
Advertisements
उत्तर
Differential equation is
(x2 – y2) dx + 2xy dy = 0
`\implies dy/dx = (-(x^2 - y^2))/(2xy)` ...(1)
Which is a Homogeneous differential equation then put y = Vx in (1)
By (1) `d/dx (Vx) = ((Vx)^2 - x^2)/(2x(Vx))`
`\implies V + x (dV)/dx = (V^2 - 1)/(2V)`
`\implies x (dV)/dx = (V^2 - 1)/(2V) - V`
`\implies x (dV)/dx = (V^2 - 1 - 2V^2)/(2V)`
`\implies x (dV)/dx = (-V^2 - 1)/(2V)`
`\implies (2V)/(V^2 + 1) dV = - dx/x`
Now integrating both sides
`int (2V)/(V^2 + 1)dV = -int dx/x`
`\implies` log (V2 + 1) = – log x + log c
`\implies log(V^2 + 1) = log(C/x)`
`\implies` V2 + 1 = `C/x`
`\implies y^2/x^2 + 1 = C/x`
`\implies` y2 + x2 = Cx.
संबंधित प्रश्न
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
Find the particular solution of the differential equation:
2y ex/y dx + (y - 2x ex/y) dy = 0 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.
`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:
x2 dy + (xy + y2) 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.
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 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:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
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 differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
`x^2. dy/dx = x^2 + xy + y^2`
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
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)`
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
The solution of the equation `dy/dx = (3x − 4y − 2)/(3x − 4y − 3)` is ______.
