Advertisements
Advertisements
Question
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
Solution
Given equation is `"dy"/"dx"` = 1 + x + y2 + xy2
⇒ `"dy"/"dx"` = 1(1 + x) + y2(1 + x)
⇒ `"dy"/"dx"` = (1 + x)(1 + y2)
⇒ `"dy"/(1 + y^2)` = (1 + x)dx
Integrating both sides, we get
`int "dy"/(1 + y^2) = int(1 + x)"d"x`
⇒ `tan^-1y = x + x^2/2 + "c"`
Put x = 0 and y = 0
We get tan–1(0) = 0 + 0 + c
⇒ c = 0
∴ tan–1y = `x + x^2/2`
⇒ y = `tan(x + x^2/2)`
Hence, the required solution is y = `tan(x + x^2/2)`.
APPEARS IN
RELATED QUESTIONS
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
x cos y dy = (xex log x + ex) dx
x cos2 y dx = y cos2 x dy
dy + (x + 1) (y + 1) dx = 0
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
Define a differential equation.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
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.
Choose the correct option from the given alternatives:
The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of
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` |
Form the differential equation from the relation x2 + 4y2 = 4b2
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
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
Solve: `("d"y)/("d"x) + 2/xy` = x2
The function y = ex is solution ______ of differential equation
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
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
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
