Advertisements
Advertisements
प्रश्न
Solve the following differential equation y2dx + (xy + x2) dy = 0
Advertisements
उत्तर
y2dx + (xy + x2) dy = 0
∴ `y^2 + (xy + x^2) ("d"y)/("d"x)` = 0
∴ `(xy + x^2) ("d"y)/("d"x)` = − y2
∴ `("d"y)/("d"x) = (-y^2)/(xy + x^2)` .....(i)
Put y = tx ......(ii)
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "t" + x "dt"/("d"x)` ......(iii)
Substituting (ii) and (iii) in (i), we get
`"t" + x "dt"/("d"x) = (-"t"^2x^2)/(x("t"x) + x^2)`
∴ `"t" + x "dt"/("d"x) = (-"t"^2x^2)/(x^2("t" + 1))`
∴ `"t" + x "dt"/("d"x) = (-"t"^2)/(1 + "t")`
∴ `x "dt"/("d"x) = (-"t"^2)/(1 + "t") - "t"`
= `(-"t"^2 - "t" - "t"^2)/(1 + "t")`
= `(-2"t"^2 - "t")/(1 + "t")`
∴ `(1 + "t")/("t"^2 + "t") "dt" = - ("d"x)/x`
Integrating on both sides, we get
`int (1 + "t")/(2"t"^2 + "t") "dt" = - int ("d"x)/x`
∴ `int ((2"t" + 1) - "t")/("t"(21"t" + 1)) "dt" = -int ("d"x)/x`
∴ `int (1/"t" - 1/(2"t" + 1)) "dt" = -int ("d"x)/x`
∴ `int 1/"t" "dt" - 1/2 int 2/(2"t" + 1) "dt" = -int ("d"x)/x`
∴ `log |"t"| - 1/2 log|2"t" + 1|` = − log|x| + log |c|
∴ `log |y/x| - 1/2 log|2(y/x) + 1|` = − log|x| + log |c|
∴ `log |y| - log |x| - 1/2 log|(2y + x)/x|` = − log|x| + log |c|
∴ `1/2 log|y^2| - 1/2 log|(2y + x)/x|` = log |c|
∴ `1/2 log|y^2/((2y + x)/x)|` = log |c|
∴ `1/2 log|(xy^2)/(2y + x)|` = log |c|
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Show that y = AeBx is a solution of the differential equation
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
xy dy = (y − 1) (x + 1) dx
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
(x + y) (dx − dy) = dx + dy
(x2 − y2) dx − 2xy dy = 0
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
(y2 − 2xy) dx = (x2 − 2xy) dy
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
`(x + y) dy/dx = 1`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Solve
`dy/dx + 2/ x y = x^2`
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
Solve the differential equation
`x + y dy/dx` = x2 + y2
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
