Advertisements
Advertisements
Question
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Advertisements
Solution
y2 dx + (xy + x2 ) dy = 0
∴ (xy + x2 ) dy = - y2 dx
∴`dy/dx = (-y^2)/(xy+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 = (-t^2x^2)/(x.tx+x^2)`
∴`t + x dt/dx = (-t^2x^2)/(tx^2+x^2)`
∴ `t + x dt/dx = (-t^2x^2)/(x^2(t+1)`
∴ ` x dt/dx = (-t^2)/(t+1)-t`
∴ ` x dt/dx = (-t^2-t^2-t)/(t+1)`
∴ ` x dt/dx = (-(2t^2+t))/(t+1)`
∴ `(t+1)/(2t^2+t)dt = - 1/x dx`
Integrating on both sides, we get
`int (t+1)/(2t^2+t)dt = -int1/xdx`
∴`int (2t + 1 - t)/(t(2t+1)) dt = - int1/xdx`
∴`int1/tdt-int 1/ (2t+1) dt = -int1/ x dx`
∴ log | t | - `1/2` log |2t + 1| = - log |x| + log |c|
∴ 2log| t | - log |2t + 1| = - 2log |x| + 2 log |c|
∴ `2log |y/x | -log |(2y)/x+ 1 |= - 2log |x| + 2 log |c|`
∴ 2log |y| - 2log |x| - log |2y + x| + log |x|
= -2log |x| + 2log |c|
∴ log |y2 | + log |x| = log |c2 | + log |2y + x|
∴ log |y2 x| = log | c2 (x + 2y)|
∴ xy 2 = c2 (x + 2y)
APPEARS IN
RELATED QUESTIONS
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
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 \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
(sin x + cos x) dy + (cos x − sin x) dx = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
dy + (x + 1) (y + 1) dx = 0
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
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?
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
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
Solve the differential equation:
dr = a r dθ − θ dr
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
Solve the differential equation
`y (dy)/(dx) + x` = 0
