Advertisements
Advertisements
Question
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Advertisements
Solution
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
∴ `("d"y)/("d"x) = 1 + y/x - (y/x)^2` .....(i)
Put `y/x` = t .....(ii)
∴ y = tx
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "t" + x ("dt")/("d"x)` .....(iii)
Substituting (ii) and (iii) in (i), we get
`"t" + x "dt"/("d"x)` = 1 + t − t2
∴ `x "dt"/("d"x)` = 1 − t2
∴ `"dt"/(1 - "t"^2) = ("d"x)/x`
Integrating on both sides, we get
`int "dt"/(1 - "t"^2) = int ("d"x)/x`
∴ `1/2 log|(1 + t)/(1 - t)|` = log |x| + log |c1|
∴ `log |(1 + y/x)/(1 - y/x)|` = 2log |x| + 2log |c1|
∴ `log|(x + y)/(x - y)|` = log |x2| + log |c12|
∴ `log|(x + y)/(x - y)|` = log |c1x2|
∴ `(x + y)/(x - y)` = c12x2
∴ `(x + y)/(x - y)` = cx2, where c = c12
RELATED QUESTIONS
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
tan y dx + sec2 y tan x dy = 0
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
2xy dx + (x2 + 2y2) dy = 0
(x + 2y) dx − (2x − y) dy = 0
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
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.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
Solve
`dy/dx + 2/ x y = x^2`
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x2 + y2
Which of the following defines a differential equation?
What distinguishes an ordinary differential equation from other types of differential equations?
