Advertisements
Advertisements
प्रश्न
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
Advertisements
उत्तर
According to the question,
\[\frac{dy}{dx} = x^2\]
\[\Rightarrow dy = x^2 dx\]
Integrating both sides with respect to x, we get
\[\int dy = \int x^2 dx\]
\[ \Rightarrow y = \frac{x^3}{3} + C\]
\[\text{ Since the curve passes through }\left( - 1, 1 \right), \text{ it satisfies the above equation .} \]
\[ \therefore 1 = \frac{- 1}{3} + C\]
\[ \Rightarrow C = 1 + \frac{1}{3}\]
\[ \Rightarrow C = \frac{4}{3}\]
Putting the value of C, we get
\[y = \frac{x^3}{3} + \frac{4}{3}\]
\[ \Rightarrow 3y = x^3 + 4\]
APPEARS IN
संबंधित प्रश्न
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
(y2 − 2xy) dx = (x2 − 2xy) dy
(x + 2y) dx − (2x − y) dy = 0
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
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 particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
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.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
`dy/dx + y = e ^-x`
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Solve the differential equation:
`e^(dy/dx) = x`
Solve the differential equation:
dr = a r dθ − θ dr
`xy dy/dx = x^2 + 2y^2`
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`
∴ (1) becomes `"dv"/("d"x) - 1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
