Advertisements
Advertisements
Question
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
Advertisements
Solution
We have,
\[\left( x y^2 + 2x \right) dx + \left( x^2 y + 2y \right) dy = 0\]
\[ \Rightarrow x\left( y^2 + 2 \right) dx + y\left( x^2 + 2 \right) dy = 0\]
\[ \Rightarrow x\left( y^2 + 2 \right) dx = - y\left( x^2 + 2 \right) dy\]
\[ \Rightarrow \frac{x}{\left( x^2 + 2 \right)} dx = - \frac{y}{\left( y^2 + 2 \right)} dy\]
Integrating both sides, we get
\[\int\frac{x}{x^2 + 2} dx = - \int\frac{y}{y^2 + 2} dy\]
\[ \Rightarrow \frac{1}{2}\int\frac{2x}{x^2 + 2} dx = - \frac{1}{2}\int\frac{2y}{y^2 + 2} dy\]
\[ \Rightarrow \frac{1}{2}log \left| x^2 + 2 \right| = - \frac{1}{2}log \left| y^2 + 2 \right| + log C\]
\[ \Rightarrow \frac{1}{2}log \left| x^2 + 2 \right| + \frac{1}{2}log \left| y^2 + 2 \right| = log C\]
\[ \Rightarrow log \left| x^2 + 2 \right| + log \left| y^2 + 2 \right| = 2log C\]
\[ \Rightarrow log \left( \left| x^2 + 2 \right|\left| y^2 + 2 \right| \right) = log C^2 \]
\[ \Rightarrow \left( \left| x^2 + 2 \right|\left| y^2 + 2 \right| \right) = C^2 \]
\[ \Rightarrow \left( x^2 + 2 \right)\left( y^2 + 2 \right) = K\]
\[ \Rightarrow y^2 + 2 = \frac{K}{x^2 + 2}\]
APPEARS IN
RELATED QUESTIONS
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
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
(ey + 1) cos x dx + ey sin x dy = 0
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
`dy/dx + 2xy = x`
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
x2y dx – (x3 + y3) dy = 0
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`
∴ (1) becomes `"dv"/("d"x) - 1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
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 ______.
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
