Advertisements
Advertisements
प्रश्न
Which of the following is a homogeneous differential equation?
विकल्प
(4x2 + 6y + 5) dy – (3y2 + 2x + 4) dx = 0
(xy) dx – (x3 + y3) dy = 0
(x3 + 2y2) dx + 2xy dy = 0
y2 dx + (x2 – xy – y2) dy = 0
Advertisements
उत्तर
y2 dx + (x2 – xy – y2) dy = 0
Explanation:
Here y2 dx + (x2 – xy – y2) dy = 0
⇒ `dy/dx = - y^2/(x^2 - xy - y^2) = y^2/(y^2 + xy - x^2)`
Now, `f (x, y) = y^2/ (y^2 + xy - x^2)`
∴ `f lambda x, lambda y = (lambda^2 y^2)/(lambda^2 y^2 + (lambda x) (lambda y) - lambda^2x^2)`
`= lambda^0 (y^2/(y^2 + xy - x^2))`
`= lambda^0 f (x, y).`
∴ f (x, y) is homogeneous function of degree zero.
संबंधित 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.
Show that the given differential equation is homogeneous and solve them.
`x^2 dy/dx = x^2 - 2y^2 + xy`
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`
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.
(2x2 y + y3) dx + (xy2 − 3x3) dy = 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:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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 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 dx + 2y dx = 0, when x = 2, y = 1
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`.
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
After using \[y=vx\] and writing the right-hand side as \[g(v)\], which separable form is obtained?
For the substitution \[x=vy\], which differentiated form is correct?
Which substitutions are chosen for a homogeneous differential equation?
What form should the equation be reduced to before integration?
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?
From \[v+x\frac{dv}{dx}=\frac{v\cos v+1}{\cos v}\], what is \[x\frac{dv}{dx}\]?
Which separable equation follows from \[x\frac{dv}{dx}=\frac{1}{\cos v}\]?
