Advertisements
Advertisements
Question
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Advertisements
Solution
`dy /dx +(x-2 y)/ (2x- y)= 0` ....(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-2tx)/(2x-tx) = 0`
∴`x dt/dx +t + (1-2t)/2-t = 0`
∴`x dt/dx + (2t - t^2+1-2t)/2-t = 0`
∴`x dt/dx + (1-t^2)/(2-t )= 0`
∴ `x dt/dx = - (1-t^2)/(2-t )`
∴ = `(2-t)/(1-t^2)dt = dx/x`
∴ `(2-t)/(t^2-1)dt = dx/x`
Integrating on both sides, we get
`int (2-t)/(t^2-1) dt = int dx/x`
∴ `int (2-t)/((t+1)(t-1)) dt = int dx/x`
Let `2-t/((t+1)(t-1)) = A/(t+1)+ B/(t-1)`
∴ 2 - t = A(t -1) + B(t + 1)
Putting t = 1, we get
∴ 2 -1 = A(1 -1) + B(1 + 1)
∴ B = `1 /2`
Putting t = -1, we get
2 -(-1) = A(-1 -1) + B(-1 + 1)
∴ A = `(-3)/2`
∴ `int(-3/2)/(t+1)dt +int(1/2)/(t-1) dt = intdx/x`
∴`(-3)/2 int 1/(t+1)dt + 1/2int 1/(t-1) dt = int dx/x`
∴`(-3)/2 log|t+1| + 1/2 log |t-1| = log |x| + log |c_1|`
∴ `-3 log |(y+x)/x| + log|(y-x)/x| = 2log |x| + 2 log |c_1|`
∴ -3 log |y+x| + 3 log |x| + log | y -x| - log |x|
= 2 log |x| + 2 log |c1|
∴ log |y - x| = 3 log |y+x|+ 2 log |c1|
∴ log |y- x |= log |( y+ x )3|+ log | c12|
∴ log | y - x| = log | c12 ( x+y)3|
∴ (y - x) = c(x + y) 3 … |c12 c|
Notes
Answer given in the textbook is `log |(x+y)/(x-y)| - 1/2 log | x^2 - y^2| + 2 log x = log c.`
However, as per our calculation it is ‘(y -x) = c(x+y)3.
APPEARS IN
RELATED QUESTIONS
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
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).
(x2 − y2) dx − 2xy dy = 0
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
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.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
Solve the following differential equation.
`(x + y) dy/dx = 1`
y2 dx + (xy + x2)dy = 0
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Solve the following differential equation y2dx + (xy + x2) dy = 0
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
