Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Advertisements
उत्तर
(x2 − y2 ) dx + 2xy dy = 0
∴ 2xy dy = (y2 - x2) dx
∴ `dy/dx = (y^2 - x^2)/(2xy) ......(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 = (t^2 x^2-x^2)/(2tx^2)`
∴ `x dt/dx = (t^2 - 1)/(2t )- t = (-(1+t^2))/(2t)`
∴ `2t/(1+t^2) dt = - dx/x`
Integrating on both sides, we get
`int 2t/(1+t^2) dt = - int dx/x`
∴ log |1 + t2| = -log |x| + log |c|
∴`log | 1+y^2/x^2| = log |c/x|`
∴ `(x^2 + y^2)/x^2 = c/x`
∴ x2 + y2 = cx
APPEARS IN
संबंधित प्रश्न
Show that y = AeBx is a solution of the differential equation
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
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?
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The solution of the differential equation y1 y3 = y22 is
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
Solve
`dy/dx + 2/ x y = x^2`
y dx – x dy + log x dx = 0
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
