Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`"dy"/"dx" + "y"/"x" = "x"^3 - 3`
उत्तर
`"dy"/"dx" + "y"/"x" = "x"^3 - 3` ...(1)
This is the linear differential equation of the form
`"dy"/"dx" + "P" * "y" = "Q"`, where P = `1/"x"` and Q = `"x"^3 - 3`
∴ I.F. = `"e"^(int "Pdx") = "e"^(int 1/"x" "dx")`
`= "e"^(log "x")` = x
∴ the solution of (1) is given by
y(I.F.) = ∫ Q. (I.F.)dx + c1
∴ `"y" * "x" = int ("x"^3 - 3)"x" "dx" + "c"_1`
∴ `"xy" = int ("x"^4 - 3"x") "dx" + "c"_1`
∴ `"xy" = "x"^5/5 - 3 * "x"^2/2 + "c"_1`
∴ `"x"^2/5 - "3x"^2/2 - "xy" = "c"`, where c = - c1
∴ This is the general 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:
`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`
For the differential equation, find the general solution:
`x dy/dx + 2y= x^2 log x`
For the differential equation, find the general solution:
`x dy/dx + y - x + xy cot x = 0(x != 0)`
For the differential equation given, find a particular solution satisfying the given condition:
`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`
Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
Find the general solution of the differential equation `dy/dx - y = sin x`
Solve the differential equation `x dy/dx + y = x cos x + sin x`, given that y = 1 when `x = pi/2`
dx + xdy = e−y sec2 y dy
Solve the differential equation \[\left( x + 2 y^2 \right)\frac{dy}{dx} = y\], given that when x = 2, y = 1.
Find the particular solution of the differential equation \[\frac{dx}{dy} + x \cot y = 2y + y^2 \cot y, y ≠ 0\] given that x = 0 when \[y = \frac{\pi}{2}\].
Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .
Solve the following differential equation:
`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`
Solve the following differential equation:
`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`
Solve the following differential equation:
`("x + y") "dy"/"dx" = 1`
Solve the following differential equation:
`("x + a")"dy"/"dx" - 3"y" = ("x + a")^5`
Solve the following differential equation dr + (2r cot θ + sin 2θ) dθ = 0.
Solve the following differential equation:
`(1 - "x"^2) "dy"/"dx" + "2xy" = "x"(1 - "x"^2)^(1/2)`
Find the equation of the curve passing through the point `(3/sqrt2, sqrt2)` having a slope of the tangent to the curve at any point (x, y) is -`"4x"/"9y"`.
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 `(dy)/(dx) + y` = e–x is ______.
Find the general solution of the equation `("d"y)/("d"x) - y` = 2x.
Solution: The equation `("d"y)/("d"x) - y` = 2x
is of the form `("d"y)/("d"x) + "P"y` = Q
where P = `square` and Q = `square`
∴ I.F. = `"e"^(int-"d"x)` = e–x
∴ the solution of the linear differential equation is
ye–x = `int 2x*"e"^-x "d"x + "c"`
∴ ye–x = `2int x*"e"^-x "d"x + "c"`
= `2{x int"e"^-x "d"x - int square "d"x* "d"/("d"x) square"d"x} + "c"`
= `2{x ("e"^-x)/(-1) - int ("e"^-x)/(-1)*1"d"x} + "c"`
∴ ye–x = `-2x*"e"^-x + 2int"e"^-x "d"x + "c"`
∴ e–xy = `-2x*"e"^-x+ 2 square + "c"`
∴ `y + square + square` = cex is the required general solution of the given differential equation
The integrating factor of the differential equation sin y `("dy"/"dx")` = cos y(1 - x cos y) is ______.
The integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.
The slope of the tangent to the curves x = 4t3 + 5, y = t2 - 3 at t = 1 is ______
Which of the following is a second order differential equation?
The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.
If y = y(x) is the solution of the differential equation, `(dy)/(dx) + 2ytanx = sinx, y(π/3)` = 0, then the maximum value of the function y (x) over R is equal to ______.
If the solution curve y = y(x) of the differential equation y2dx + (x2 – xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = `sqrt(3) x` at the point `(α, sqrt(3) α)`, then value of `log_e (sqrt(3)α)` is equal to ______.
If the slope of the tangent at (x, y) to a curve passing through `(1, π/4)` is given by `y/x - cos^2(y/x)`, then the equation of the curve is ______.
Find the general solution of the differential equation:
`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`
