Advertisements
Advertisements
Question
Solve the following differential equation y2dx + (xy + x2) dy = 0
Advertisements
Solution
y2dx + (xy + x2) dy = 0
∴ `y^2 + (xy + x^2) ("d"y)/("d"x)` = 0
∴ `(xy + x^2) ("d"y)/("d"x)` = − y2
∴ `("d"y)/("d"x) = (-y^2)/(xy + x^2)` .....(i)
Put y = tx ......(ii)
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) = (-"t"^2x^2)/(x("t"x) + x^2)`
∴ `"t" + x "dt"/("d"x) = (-"t"^2x^2)/(x^2("t" + 1))`
∴ `"t" + x "dt"/("d"x) = (-"t"^2)/(1 + "t")`
∴ `x "dt"/("d"x) = (-"t"^2)/(1 + "t") - "t"`
= `(-"t"^2 - "t" - "t"^2)/(1 + "t")`
= `(-2"t"^2 - "t")/(1 + "t")`
∴ `(1 + "t")/("t"^2 + "t") "dt" = - ("d"x)/x`
Integrating on both sides, we get
`int (1 + "t")/(2"t"^2 + "t") "dt" = - int ("d"x)/x`
∴ `int ((2"t" + 1) - "t")/("t"(21"t" + 1)) "dt" = -int ("d"x)/x`
∴ `int (1/"t" - 1/(2"t" + 1)) "dt" = -int ("d"x)/x`
∴ `int 1/"t" "dt" - 1/2 int 2/(2"t" + 1) "dt" = -int ("d"x)/x`
∴ `log |"t"| - 1/2 log|2"t" + 1|` = − log|x| + log |c|
∴ `log |y/x| - 1/2 log|2(y/x) + 1|` = − log|x| + log |c|
∴ `log |y| - log |x| - 1/2 log|(2y + x)/x|` = − log|x| + log |c|
∴ `1/2 log|y^2| - 1/2 log|(2y + x)/x|` = log |c|
∴ `1/2 log|y^2/((2y + x)/x)|` = log |c|
∴ `1/2 log|(xy^2)/(2y + x)|` = log |c|
RELATED QUESTIONS
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Show that y = AeBx is a solution of the differential equation
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
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 \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x
x cos2 y dx = y cos2 x dy
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
(y + xy) dx + (x − xy2) dy = 0
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
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:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
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.
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
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.
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The solution of the differential equation y1 y3 = y22 is
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
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 representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
`dy/dx = x^2 y + y`
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
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + 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.
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
