Advertisements
Advertisements
प्रश्न
xy (y + 1) dy = (x2 + 1) dx
Advertisements
उत्तर
We have,
\[xy\left( y + 1 \right)dy = \left( x^2 + 1 \right)dx\]
\[ \Rightarrow \left\{ y\left( y + 1 \right) \right\}dy = \frac{x^2 + 1}{x}dx\]
\[ \Rightarrow \left( y^2 + y \right)dy = \left( x + \frac{1}{x} \right)dx\]
Integrating both sides, we get
\[\int\left( y^2 + y \right)dy = \int\left( x + \frac{1}{x} \right)dx\]
\[ \Rightarrow \int y^2 dy + \int y dy = \int x dx + \int\frac{1}{x}dx\]
\[ \Rightarrow \frac{y^3}{3} + \frac{y^2}{2} = \frac{x^2}{2} + \log \left| x \right| + C\]
\[\text{ Hence,} \frac{y^3}{3} + \frac{y^2}{2} = \frac{x^2}{2} + \log \left| x \right| +\text{ C is the required solution }. \]
APPEARS IN
संबंधित प्रश्न
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
(x + y) (dx − dy) = dx + dy
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
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:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
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.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
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=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Form the differential equation from the relation x2 + 4y2 = 4b2
Solve the following differential equation.
`dy/dx = x^2 y + y`
x2y dx – (x3 + y3) dy = 0
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Solve the differential equation `"dy"/"dx" + 2xy` = y
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
Which of the following defines a differential equation?
What distinguishes an ordinary differential equation from other types of differential equations?
Why is the equation \[x\frac{dy}{dx} + y = 0\] classified as a differential equation?
