Advertisements
Advertisements
प्रश्न
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
Advertisements
उत्तर
The given differential equation is
`(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = ("x"^2 + 2)`
This equation is of the form `(d"y")/(d"x") + P"y" = "Q", "where P" = (-2"x")/(1+ "x"^2) and "Q" = "x"^2 + 2`
Now, `"I.F." e^(intPd"x") = e^(int_ (-2"x")/(1+"x"^2) d"x" =e^-log(1+"x"^2) = e^log((1)/(1+"x"^2)) = (1)/(1+"x"^2)`
The general solution of the given differential equation is
`"y" xx "I.F." = int_ ("Q" xx "I.F.") d"x" + "C", "where C is an aribatry constant"`
⇒ `("y")/(1 +"x"^2) = int_ ("x"^2 + 2)/(1+ "x"^2) d"x" + "C"`
= `int_ (1 + (1)/("x"^2 + 1)) d"x" + "C"`
= `int_ d"x" + int(1)/("x"^2 + 1) d"x" + "C"`
= `"x" + tan^-1 "x" + "C"`
`"y" = (1 + "x"^2)("x" + tan^-1 "x" + "C")`
APPEARS IN
संबंधित प्रश्न
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
If y = etan x+ (log x)tan x then find dy/dx
The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
(x3 − 2y3) dx + 3x2 y dy = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
