Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Advertisements
उत्तर
`xy dy/dx = x^2 + 2y^2`
∴ `dy/dx = (x^2 + 2y^2)/(xy) …(i)`
Put y = tx ...(ii)
Differentiating w.r.t. x, we get
`dy/dx = t + x dt/dx` ...(iii)
Substituting (ii) and (iii) in (i), we get
`t +x dt/dx = (x^2 + 2t^2 x^2)/(x(tx))`
∴`t +x dt/dx = (x^2 (1 + 2t^2))/(x^2t)`
∴ `x dt/dx = (1 + 2t^2)/t - t = (1+t^2)/t`
∴ `t / (1+t^2) dt = 1/xdx`
Integrating on both sides, we get
`1/2 int (2t)/(1+t^2) dt = int dx/x`
∴ `1/2 log |1+ t^2| = log|x| + log |c_1|`
∴ log |1 + t2 | = 2 log |x| + 2log |c1|
= log |x2| + log |c| …[logc12 = log c]
∴ log |1 + t2| = log |cx 2|
∴ 1 + t2 = cx2
∴ `1+ y^2/x^2 = cx^2`
∴ x2 + y2 = cx4
APPEARS IN
संबंधित प्रश्न
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
xy (y + 1) dy = (x2 + 1) dx
(1 − x2) dy + xy dx = xy2 dx
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
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.
The differential equation `y dy/dx + x = 0` represents family of ______.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve: `("d"y)/("d"x) + 2/xy` = x2
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
