Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Advertisements
उत्तर
`x^2 dy/dx = x^2 +xy - y^2`
∴ `dy/dx = (x^2 + xy - y^2)/x^2` …(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 + x(tx) - (tx)^2)/x^2`
∴ `t+x dt/dx = (x^2 + tx^2 - t^2x^2)/ x^2`
∴ `t+x dt/dx = 1 +t -t^2`
∴ `x dt/dx = 1 + t - t^2`
∴ `x dt/dx = 1 -t^2`
∴ `dt/(1^2- t^2) = dx/x`
Integrating on both sides, we get
`int dt/((1)^2 - (t)^2) = int dx/x`
∴ `1/(2(1)) log | (1+t)/(1-t)| = log |x| + log |c_1|` ...[`Qint dx/(a^2 - x^2) = 1/(2a) log | (a+x)/(a-x)| +c]`
∴ `log |(1+t)/(1-t)|= log |x| + log|c_1|`
∴ `log |(1+t)/(1-t)|= log |c_1^2x^2|`
∴ `(1+(y/x))/(1-(y/x)) = c_1^2x^2`
∴ `(x+y)/(x-y) = cx^2 … [c_1^2 = c]`
APPEARS IN
संबंधित प्रश्न
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
(x2 − y2) dx − 2xy dy = 0
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 curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
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.
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.
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
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
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.
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.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
