Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Advertisements
उत्तर
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
\[ \Rightarrow \left( 1 + y^2 \right) \tan^{- 1} xdx = - 2y\left( 1 + x^2 \right)dy\]
\[ \Rightarrow \frac{\tan^{- 1} x}{\left( 1 + x^2 \right)}dx = - \frac{2y}{\left( 1 + y^2 \right)}dy\]
\[ \Rightarrow \int\frac{\tan^{- 1} x}{\left( 1 + x^2 \right)}dx = - \int\frac{2y}{\left( 1 + y^2 \right)}dy\]
\[ \Rightarrow \frac{\left( \tan^{- 1} x \right)^2}{2} = - \log\left| 1 + y^2 \right| + C\]
\[ \Rightarrow \frac{\left( \tan^{- 1} x \right)^2}{2} + \log\left| 1 + y^2 \right| = C\]
APPEARS IN
संबंधित प्रश्न
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
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 y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
(1 − x2) dy + xy dx = xy2 dx
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
(x2 − y2) dx − 2xy dy = 0
(y2 − 2xy) dx = (x2 − 2xy) dy
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
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.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The solution of the differential equation y1 y3 = y22 is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve:
(x + y) dy = a2 dx
y2 dx + (xy + x2)dy = 0
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Which of the following is an example of an ordinary differential equation?
