Advertisements
Advertisements
Question
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.
Solution
Let A (x, y) be any point on the curve y = f(x).
Then the slope of the tangent to the curve at point A is `"dy"/"dx"`.
According to the given condition
`"dy"/"dx"` = x + xy
∴ `"dy"/"dx"` - xy = x .....(1)
This is the linear differential equation of the form
`"dy"/"dx" + "P" "y" = "Q"`, where P = - x and Q = x
∴ I.F. = `"e"^(int "P dx") = "e"^(int -"x" "dx") = "e"^(-"x"^2/2)`
∴ the solution of (1) is given by
`"y"*("I.F.") = int "Q" * ("I.F.") "dx" + "c"`
∴ `"y" * "e"^(-"x"^2/2) = = int "x" * "e"^(-"x"^2/2) "dx" + "c"`
∴ `"e"^(-"x"^2/2) * "y" = int "x" * "e"^(-"x"^2/2) "dx" + "c"`
∴ Put `- "x"^2/2 = "t"`
∴ - x dx = dt
∴ x dx = - dt
∴ `"e"^(-"x"^2/2) * "y" = int "e"^"t" * (- "dt") + "c"`
∴ `"e"^(-"x"^2/2) * "y" = - int "e"^"t" * "dt" + "c"`
∴ `"e"^(-"x"^2/2) * "y" = - "e"^"t" + "c"`
∴ `"e"^(-"x"^2/2) * "y" = - "e"^(-"x"^2/2) + "c"`
∴ y = - 1 + `"ce"^("x"^2/2)`
∴ 1 + y = `"ce"^("x"^2/2)` .....(2)
This is the general equation of the curve.
But the required curve is passing through the point (0, 1).
∴ by putting x = 0 and y = 1 in (2), we get
1 + 1 = c
∴ c = 2
∴ from (2), the equation of the required curve is
1 + y = `2"e"^("x"^2/2)`.
APPEARS IN
RELATED QUESTIONS
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 + 2y = sin x`
For the differential equation, find the general solution:
`x log x dy/dx + y= 2/x log x`
For the differential equation, find the general solution:
(1 + x2) dy + 2xy dx = cot x dx (x ≠ 0)
For the differential equation, find the general solution:
y dx + (x – y2) dy = 0
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 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 the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.
The integrating factor of the differential equation.
`(1 - y^2) dx/dy + yx = ay(-1 < y < 1)` is ______.
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`
\[\frac{dy}{dx}\] + y cos x = sin x cos x
Solve the differential equation \[\left( x + 2 y^2 \right)\frac{dy}{dx} = y\], given that when x = 2, y = 1.
Find the general solution of the differential equation \[\frac{dy}{dx} - y = \cos x\]
Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .
Solve the differential equation: (1 +x2 ) dy + 2xy dx = cot x dx
Solve the following differential equation:
`"dy"/"dx" + "y"/"x" = "x"^3 - 3`
Solve the following differential equation:
y dx + (x - y2) dy = 0
Solve the following differential equation:
`(1 - "x"^2) "dy"/"dx" + "2xy" = "x"(1 - "x"^2)^(1/2)`
Solve the following differential equation:
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
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.
The integrating factor of the differential equation sin y `("dy"/"dx")` = cos y(1 - x cos y) is ______.
The slope of the tangent to the curves x = 4t3 + 5, y = t2 - 3 at t = 1 is ______
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 ______.
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 ______.
Let the solution curve y = y(x) of the differential equation (4 + x2) dy – 2x (x2 + 3y + 4) dx = 0 pass through the origin. Then y (2) 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 ______.
Solve:
`xsinx dy/dx + (xcosx + sinx)y` = sin x
The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.
