Advertisements
Advertisements
प्रश्न
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Advertisements
उत्तर
The given differential equation is `y + "d"/("d"x) (xy) = x(sinx + logx)`
⇒ `y + x * ("d"y)/("d"x) + y = x(sinx + logx)`
⇒ `x ("d"y)/("d"x) = x(sinx + logx) - 2y`
⇒ `("d"y)/("d"x) = (sinx + logx) - (2y)/x`
⇒ `("d"y)/("d"x) + 2x y = (sinx + logx)`
Here, P = `2/x` and Q = `(sinx + log x)`
Integrating factor I.F. = `"e"^(intPdx)`
= `"e"^(int 2/x dx)`
= `"e"^(2logx)`
= `"e"^(log x^2)`
= x2
∴ Solution is `y xx "I"."F". = int "Q"."I"."F". "d"x + "c"`
⇒ `y . x^2 = int (sinx + logx)x^2 "d"x + "c"` ....(1)
Let I = `int (sinx + logx)x^2 "d"x`
= `int_"I"x^2 sinx "d"x + int_"iII"^(x^2) log x "d"x`
= `[x^2 . int sinx "d"x - int("D"(x^2) . int sinx "d"x)"d"x] + [logx . intsinx "d"x - int ("D"(logx) . intx^2 "d"x)"d"x]`
= `[x^2(-cosx) -2 int - x cosx "d"x] + [logx . x^3/3 - int 1/x * x^3/3 "d"x]`
= `[-x^2 cosx + 2(xsinx - int1 .sinx "d"x)] + [x^3/3 log x - 1/3 int x^2 "d"x]`
= `-x^2cosx + 2x sinx + 2cosx + x^3/3 log x - 1/9 x^3`
Now from equation (1) we get,
`y . x^2 = -x^2 cosx + 2x sinx + 2cosx + x^3/3 log x - 1/9 x^3 + "c"`
∴ y = `-cosx + (2sinx)/x + (2cosx)/x^2 + (xlogx)/3 - 1/9 x + "c" .x^-2`
Hence, the required solution is `-cosx + (2sinx)/x + (2cosx)/x^2 + (xlogx)/3 - 1/9 x + "c" .x^-2`
APPEARS IN
संबंधित प्रश्न
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
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`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\], then
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
The number of arbitrary constants in the general solution of differential equation of fourth order is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
\[\frac{dy}{dx} + 1 = e^{x + y}\]
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Which of the following differential equations has `y = x` as one of its particular solution?
