Advertisements
Advertisements
Question
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
Advertisements
Solution
(x2 + 3xy + y2)dx - x2 dy = 0
∴ x2 dy = (x2 + 3xy + y2)dx
∴ `"dy"/"dx" = ("x"^2 + 3"xy" + "y"^2)/"x"^2` ....(1)
Put y = vx
∴ `"dy"/"dx" = "v + x" "dv"/"dx"`
∴ (1) becomes, `"v + x" "dv"/"dx" = ("x"^2 + 3"x" * "vx" + "v"^2"x"^2)/"x"^2`
∴ `"v + x" "dv"/"dx" = 1 + 3"v" + "v"^2`
`"x" "dv"/"dx" = "v"^2 + 2"v" + 1 = ("v + 1")^2`
∴ `1/("v + 1")^2 "dv" = 1/"x" "dx"`
Integrating, we get
`int ("v + 1")^-2 "dv" = int 1/"x" "dx"`
∴ `("v + 1")^-1/-1 = log |"x"| + "c"_1`
∴ `- 1/("v + 1") = log |"x"| + "c"_1`
∴ `- 1/("y"/"x" + 1) = log |"x"| + "c"_1`
∴ `- "x"/("y + x") = log |"x"| + "c"_1`
∴ `log |"x"| + "x"/("x + y") = - "c"_1`
∴ `log |"x"| + "x"/("x + y") = "c"`, where c = - c1
This is the general solution.
APPEARS IN
RELATED QUESTIONS
Show that the differential equation 2yx/y dx + (y − 2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.
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.
`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/dx - y + x sin (y/x) = 0`
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:
(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`
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
(2x2 y + y3) dx + (xy2 − 3x3) dy = 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:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
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, \[y = \frac{\pi}{4}\]
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:
`x * dy/dx - y + x * sin(y/x) = 0`
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 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`
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
Solve the following differential equation:
(x2 – y2)dx + 2xy 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`.
Which of the following is not a homogeneous function of x and y.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
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 ______.
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)
A function \[F(x,y)\] is homogeneous of degree \[n\] when which condition holds?
Which substitutions are chosen for a homogeneous differential equation?
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?
After putting \[y=vx\] in \[\frac{dy}{dx}=\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\], which equation results?
Which separable equation follows from \[x\frac{dv}{dx}=\frac{1}{\cos v}\]?
Which sequence correctly states the essential actions for obtaining the final answer of a homogeneous differential equation?
