English

Find the general solution of the equation dydx-y = 2x. Solution: The equation dydx-y = 2x is of the form dydx+Py = Q where P = □ and Q = □ ∴ I.F. = e∫-dx = e–x ∴ the solution of the linear d - Mathematics and Statistics

Advertisements
Advertisements

Question

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

Fill in the Blanks
Sum
Advertisements

Solution

The equation `("d"y)/("d"x) - y` = 2x

is of the form `("d"y)/("d"x) + "P"y` = Q

where P = – 1 and Q = 2x

∴ 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 "e"^-x  "d"x* "d"/("d"x) x "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 ("e"^-x)/(-1) + "c"`

∴ y + 2x + 2 = cex is the required general solution of the given differential equation

shaalaa.com
  Is there an error in this question or solution?
Chapter 1.8: Differential Equation and Applications - Q.6

APPEARS IN

SCERT Maharashtra Mathematics and Statistics (Commerce) [English] 12 Standard HSC
Chapter 1.8 Differential Equation and Applications
Q.6 | Q 1

RELATED QUESTIONS

For the differential equation, find the general solution:

`x dy/dx +  2y= x^2 log x`


For the differential equation, find the general solution:

y dx + (x – y2) dy = 0


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 ______.


The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?


Find the general solution of the differential equation `dy/dx - y = sin x`


\[\frac{dy}{dx}\] = y tan x − 2 sin x


\[\left( \sin x \right)\frac{dy}{dx} + y \cos x = 2 \sin^2 x \cos x\]

\[\frac{dy}{dx} - y = x e^x\]

\[\frac{dy}{dx} + 2y = x e^{4x}\]

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:

`("x" + 2"y"^3) "dy"/"dx" = "y"`


Solve the following differential equation:

`"dy"/"dx" + "y" * sec "x" = tan "x"`


Solve the following differential equation:

`("x + y") "dy"/"dx" = 1`


Solve the following differential equation:

dr + (2r cotθ + sin2θ)dθ = 0


Solve the following differential equation:

`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`


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 ______.


`(x + 2y^3 ) dy/dx = y`


Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 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 ______.


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 ______.


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 ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×