Advertisements
Advertisements
Question
dy + (x + 1) (y + 1) dx = 0
Advertisements
Solution
We have,
\[dy + \left( x + 1 \right)\left( y + 1 \right) dx = 0\]
\[ \Rightarrow dy = - \left( x + 1 \right)\left( y + 1 \right) dx\]
\[ \Rightarrow \frac{1}{y + 1}dy = - \left( x + 1 \right) dx\]
Integrating both sides, we get
\[\int\frac{1}{y + 1}dy = - \int\left( x + 1 \right) dx\]
\[ \Rightarrow \log \left| y + 1 \right| = - \frac{x^2}{2} - x + C\]
\[ \Rightarrow \log \left| y + 1 \right| + \frac{x^2}{2} + x = C\]
\[\text{ Hence, }\log \left| y + 1 \right| + \frac{x^2}{2} + x =\text{ C is the required solution . }\]
APPEARS IN
RELATED QUESTIONS
Verify that y = cx + 2c2 is a solution of the differential equation
C' (x) = 2 + 0.15 x ; C(0) = 100
(1 − x2) dy + xy dx = xy2 dx
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
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).
A population grows at the rate of 5% per year. How long does it take for the population to double?
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 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).
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.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\] are rectangular hyperbola.
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).
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
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 the differential equation:
`e^(dy/dx) = x`
Solve the differential equation:
dr = a r dθ − θ dr
`dy/dx = log x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
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.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation
`x + y dy/dx` = x2 + y2
Why is the equation \[x\frac{dy}{dx} + y = 0\] classified as a differential equation?
