Advertisements
Advertisements
Question
Solve the following differential equation:
`"x"^2 "dy"/"dx" = "x"^2 + "xy" + "y"^2`
Solution
`"x"^2 "dy"/"dx" = "x"^2 + "xy" + "y"^2`
∴ `"dy"/"dx" = ("x"^2 + "xy" + "y"^2)/"x"^2` ...(1)
Put y = vx
∴ `"dy"/"dx" = "v + x""dv"/"dx"`
∴ (1) becomes, `"v + x""dv"/"dx" = ("x"^2 + "x" * "vx" + "v"^2"x"^2)/"x"^2`
∴ `"v + x""dv"/"dx" = 1 + "v" + "v"^2`
∴ `"x" "dv"/"dx" = 1 + "v"^2`
∴ `1/(1 + "v"^2) "dv" = 1/"x" "dx"`
Integrating, we get
`int 1/(1 + "v"^2) "dv" = int 1/"x" "dx"`
∴ tan-1 v = log |x| + c
∴ tan-1 `("y"/"x") = log |"x"| + "c"`
This is the general solution.
APPEARS IN
RELATED QUESTIONS
Solve the differential equation (x2 + y2)dx- 2xydy = 0
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.
`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:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
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.
Which of the following is a homogeneous differential equation?
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.
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
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
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 0
Solve the following initial value problem:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
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:
(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 \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], 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: ` (dy)/(dx) = (x + y )/ (x - y )`
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:
`"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:
(9x + 5y) dy + (15x + 11y)dx = 0
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 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`.
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
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)`
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
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
