Advertisements
Advertisements
प्रश्न
dy + (x + 1) (y + 1) dx = 0
उत्तर
We have,
\[dy + \left( x + 1 \right)\left( y + 1 \right) dx = 0\]
\[ \Rightarrow dy = - \left( x + 1 \right)\left( y + 1 \right) dx\]
\[ \Rightarrow \frac{1}{y + 1}dy = - \left( x + 1 \right) dx\]
Integrating both sides, we get
\[\int\frac{1}{y + 1}dy = - \int\left( x + 1 \right) dx\]
\[ \Rightarrow \log \left| y + 1 \right| = - \frac{x^2}{2} - x + C\]
\[ \Rightarrow \log \left| y + 1 \right| + \frac{x^2}{2} + x = C\]
\[\text{ Hence, }\log \left| y + 1 \right| + \frac{x^2}{2} + x =\text{ C is the required solution . }\]
APPEARS IN
संबंधित प्रश्न
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
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?
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.
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.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
y2 dx + (x2 − xy + y2) dy = 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
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
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.
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation.
`dy/dx + y = e ^-x`
State whether the following is True or False:
The integrating factor of the differential equation `dy/dx - y = x` is e-x
Solve the differential equation:
`e^(dy/dx) = x`
x2y dx – (x3 + y3) dy = 0
`dy/dx = log x`
Solve the following differential equation y2dx + (xy + x2) dy = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
