Advertisements
Advertisements
प्रश्न
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 0)`
Advertisements
उत्तर
`(x + 3y^2) dy/dx = y`
or `y dx/dy = x + 3y^2`
`dx/dy - x/y = 3y`
A vertical differential equation of the form `dx/dy + Px = Q.`
Here `P = - 1/y, Q = 3y`
∴ `I.F. = e^(int P dx) = e^(- int 1/y dy) = e^(- log y) = 1/y`
Hence, the general solution of the differential equation
⇒ `x × I.F. = int Q xx I.F. dx + C`
⇒ `x xx 1/y = int 1/y (3y) dy + C`
⇒ `x/y = 3 int 1 dy + C`
⇒ `x/y = 3y + C`
⇒ x = 3y2 + Cy
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
Find the the differential equation for all the straight lines, which are at a unit distance from the origin.
For the differential equation, find the general solution:
`dy/dx + y/x = x^2`
For the differential equation, find the general solution:
`dy/dx + (sec x) y = tan x (0 <= x < pi/2)`
For the differential equation, find the general solution:
`x dy/dx + y - x + xy cot x = 0(x != 0)`
For the differential equation, find the general solution:
`(x + y) dy/dx = 1`
For the differential equation, find the general solution:
y dx + (x – y2) dy = 0
Solve the differential equation `(tan^(-1) x- y) dx = (1 + x^2) dy`
x dy = (2y + 2x4 + x2) dx
(x + tan y) dy = sin 2y dx
Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
Solve the differential equation \[\frac{dy}{dx}\] + y cot x = 2 cos x, given that y = 0 when x = \[\frac{\pi}{2}\] .
Solve the following differential equation:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]
Find the integerating factor of the differential equation `x(dy)/(dx) - 2y = 2x^2`
Find the integerating factor of the differential equation `xdy/dx - 2y = 2x^2` .
Solve the differential equation: (1 +x2 ) dy + 2xy dx = cot x dx
If f(x) = x + 1, find `"d"/"dx"("fof") ("x")`
Solve the following differential equation:
`("x" + 2"y"^3) "dy"/"dx" = "y"`
Solve the following differential equation:
`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`
Solve the following differential equation:
dr + (2r cotθ + sin2θ)dθ = 0
The curve passes through the point (0, 2). The sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at any point by 5. Find the equation of the curve.
If the slope of the tangent to the curve at each of its point is equal to the sum of abscissa and the product of the abscissa and ordinate of the point. Also, the curve passes through the point (0, 1). Find the equation of the curve.
Form the differential equation of all circles which pass through the origin and whose centers lie on X-axis.
The integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.
Which of the following is a second order differential equation?
The equation x2 + yx2 + x + y = 0 represents
State whether the following statement is true or false.
The integrating factor of the differential equation `(dy)/(dx) + y/x` = x3 is – x.
Let y = y(x), x > 1, be the solution of the differential equation `(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`, with y(2) = `(1 + e^4)/(2e^4)`. If y(3) = `(e^α + 1)/(βe^α)`, then the value of α + β is equal to ______.
Let y = y(x) be a solution curve of the differential equation (y + 1)tan2xdx + tanxdy + ydx = 0, `x∈(0, π/2)`. If `lim_(x→0^+)` xy(x) = 1, then the value of `y(π/2)` is ______.
Let y = f(x) be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If f(x) satisfies xf'(x) = x2 + f(x) – 2, then the area bounded by f(x) with x-axis between ordinates x = 0 and x = 3 is equal to ______.
If sin x is the integrating factor (IF) of the linear differential equation `dy/dx + Py` = Q then P is ______.
Find the general solution of the differential equation:
`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`
If sec x + tan x is the integrating factor of `dy/dx + Py` = Q, then value of P is ______.
Solve:
`xsinx dy/dx + (xcosx + sinx)y` = sin x
