Advertisements
Advertisements
Question
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
Advertisements
Solution
(x − y2 x)dx − (y + x2 y) dy = 0, when x = 2, y = 0
∴ x(1- y2) dx = y(1 + x2 ) dy
∴ `(xdx)/(1+x^2) = (ydy)/(1-y^2)`
Integrating on both sides, we get
`int( 2x)/(1+x^2) dx = int(2y)/(1-y^2 )dy`
∴ `int( 2x)/(1+x^2) dx = - int(-2y)/(1-y^2 )dy`
∴ `log | 1 + x^2| = -log| 1-y^2| + log |c|`
∴ `log |1 + x^2 | = log |c /(1-y^2)|`
∴ (1 + x 2) ( 1 - y2 ) = c …(i)
When x = 2, y = 0, we have
(1 + 4) (1 - 0) = c
∴ c = 5
Substituting c = 5 in (i),we get
(1 + x2) ( 1-y2 ) = 5,
which is the required particular solution.
APPEARS IN
RELATED QUESTIONS
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
x cos2 y dx = y cos2 x dy
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the differential equation:
`e^(dy/dx) = x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c1
∴ e–2y = – 2sin x – 2c1
∴ `square` = c, where c = – 2c1
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
