Advertisements
Advertisements
प्रश्न
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
उत्तर
We have,
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x\left( x^2 - 1 \right)}\]
\[ \Rightarrow dy = \left\{ \frac{1}{x\left( x^2 - 1 \right)} \right\}dx\]
Integrating both sides, we get
\[\int dy = \int\left\{ \frac{1}{x\left( x^2 - 1 \right)} \right\}dx\]
\[ \Rightarrow y = \int\left\{ \frac{1}{x\left( x^2 - 1 \right)} \right\}dx + C\]
\[ \Rightarrow y = \int\left\{ \frac{1}{x\left( x + 1 \right)\left( x - 1 \right)} \right\}dx + C . . . . . . . . \left( 1 \right)\]
\[\text{Let }\frac{1}{x\left( x + 1 \right)\left( x - 1 \right)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}\]
\[ \Rightarrow 1 = A\left( x + 1 \right)\left( x - 1 \right) + Bx\left( x - 1 \right) + Cx\left( x + 1 \right)\]
\[ \Rightarrow 1 = A\left( x^2 - 1 \right) + B\left( x^2 - x \right) + C\left( x^2 + x \right)\]
\[ \Rightarrow 1 = x^2 \left( A + B + C \right) + x\left( - B + C \right) - A\]
Comparing both sides, we get
\[ - A = 1 . . . . . . . . . (2)\]
\[ - B + C = 0 . . . . . . . . .(3)\]
\[A + B + C = 0 . . . . . . . . (4)\]
Solving (2), (3) and (4), we get
\[A = - 1\]
\[B = \frac{1}{2}\]
\[C = \frac{1}{2}\]
\[ \therefore \frac{1}{x\left( x + 1 \right)\left( x - 1 \right)} = \frac{- 1}{x} + \frac{1}{2\left( x + 1 \right)} + \frac{1}{2\left( x - 1 \right)}\]
Now, (1) becomes
\[y = \int\left\{ \frac{- 1}{x} + \frac{1}{2\left( x + 1 \right)} + \frac{1}{2\left( x - 1 \right)} \right\}dx + C\]
\[ \Rightarrow y = - \int\frac{1}{x}dx + \frac{1}{2}\int\frac{1}{x - 1}dx + \frac{1}{2}\int\frac{1}{x - 1}dx\]
\[ \Rightarrow y = - \log \left| x \right| + \frac{1}{2}\log \left| x - 1 \right| + \frac{1}{2}\log \left| x + 1 \right| + C\]
\[ \Rightarrow y = \frac{1}{2}\log \left| x - 1 \right| + \frac{1}{2}\log \left| x + 1 \right| - \log \left| x \right| + C\]
Given:- `y(2) = 0`
\[ \therefore 0 = \frac{1}{2}\log \left| 2 - 1 \right| + \frac{1}{2}\log \left| 2 + 1 \right| - \log \left| 2 \right| + C\]
\[ \Rightarrow C = \log \left| 2 \right| - \frac{1}{2}\log \left| 3 \right|\]
Substituting the value of `C`, we get
\[y = \frac{1}{2}\log \left| x - 1 \right| + \frac{1}{2}\log \left| x + 1 \right| - \log \left| x \right| + \log \left| 2 \right| - \frac{1}{2}\log \left| 3 \right|\]
\[ \Rightarrow 2y = \log \left| x - 1 \right| + \log \left| x + 1 \right| - 2\log \left| x \right| + 2\log \left| 2 \right| - \log \left| 3 \right|\]
\[ \Rightarrow 2y = \log \left| x - 1 \right| + \log \left| x + 1 \right| - \log \left| x^2 \right| + \log \left| 4 \right| - \log \left| 3 \right|\]
\[ \Rightarrow 2y = \log\frac{\left( x - 1 \right)\left( x + 1 \right)}{x^2} - \left( \log\left| 3 \right| - \log\left| 4 \right| \right)\]
\[ \Rightarrow y = \frac{1}{2}\log\frac{\left( x^2 - 1 \right)}{x^2} - \frac{1}{2}\log \left( \frac{3}{4} \right)\]
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
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`
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=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:
x + y = tan–1y : y2 y′ + y2 + 1 = 0
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
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 `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
If y = etan x+ (log x)tan x then find dy/dx
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.
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
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
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
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 particular solution of a differential equation of third order is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
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:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
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 a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 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.`
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 number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve: `2(y + 3) - xy "dy"/"dx"` = 0, given that y(1) = – 2.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
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) = "e"^(x - y) + x^2 "e"^-y` is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of differential equation coty dx = xdy is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
