Advertisements
Advertisements
Question
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Advertisements
Solution
(x2 – yx2)dy + (y2 + xy2)dx = 0
∴ x2(1 – y) dy + y2(1 + x) dx = 0
∴ x2(1 – y) dy = – y2(1 + x) dx
∴ `((1 - y)/y^2) "d"y = -((1 + x)/x^2) "d"x`
Integrating on both sides, we get
`int ((1 - y)/y^2) "d"y = -int((1 + x)/x^2) "d"x`
∴ `int 1/y^2 "d"y -int 1/y "d"y = -int 1/x^2 "d"x - int 1/x "d"x`
∴ `y^(-1)/(-1) - log|y| = (x^(-1)/(-1)) - log|x| + "c"`
∴ `- 1/y - log|y| = 1/x - log|x| + "c"`
∴ log |x| − log |y| = `1/x + 1/y + "c"`
APPEARS IN
RELATED QUESTIONS
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
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 = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
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 \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(1 + x2) dy = xy dx
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
(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 \[x=\sqrt{2}\]
(x + y) (dx − dy) = dx + dy
x2 dy + y (x + y) dx = 0
(y2 − 2xy) dx = (x2 − 2xy) dy
3x2 dy = (3xy + y2) dx
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
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.
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 x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
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 \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation y1 y3 = y22 is
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation satisfied by ax2 + by2 = 1 is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
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
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
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` |
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 each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution 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.
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.
Solve:
(x + y) dy = a2 dx
Solve
`dy/dx + 2/ x y = x^2`
y2 dx + (xy + x2)dy = 0
`dy/dx = log x`
Solve the differential equation xdx + 2ydy = 0
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
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`
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
