Advertisements
Advertisements
प्रश्न
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
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
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The number of arbitrary constants in the general solution of differential equation of fourth order is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + 5y = \cos 4x\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
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:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
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.`
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
