Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
`2ye^(x/y)dx+(y-2xe^(x/y))dy=0`
`=>dx/dy=(2xe^(x/y-y))/(2ye^(x/y))`
Given differential equation is a homogeneous differential equation.
∴ Put x = vy
`dx/dy=v+y (dv)/dy`
`v+y(dv)/dy=(2ve^v-1)/(2e^v)`
`=>y(dv)/dy=(2ve^v-1)/(2e^v)-v`
`=>y(dv)/dy=-1/(2e^v)`
`=>2e^vdv=-1/ydy`
Integrating on both the sides
`=>2inte^vdv=-int1/ydy`
`=>2e^v=-log|y|+logC`
`=>2e^v=log|c/y|`
`=>2e^(x/y)=log|c/y|`
Given that at x = 0, y = 1
`2e^0= log|c/1|`
⇒ C = e2
`:.2e^(x/y)=log""e^2/y`
`=>logy=-2e^(x/y)+2`
`=>y=e^2-2e^(x/y)`
संबंधित प्रश्न
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:
x2 dy + (xy + y2) dx = 0; y = 1 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.
Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy, where C is parameter
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(x2 + 3xy + y2) dx − x2 dy = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
Solve the following initial value problem:
(xy − y2) dx − x2 dy = 0, y(1) = 1
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:
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:
`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.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
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.
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 ______.
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
