Advertisements
Advertisements
Question
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Advertisements
Solution
dr + (2r)dθ= 8dθ
`(dr)/(dθ)` + 2r = 8
The given equation is of the form
`(dr)/(dθ) + Pr = Q`
where, P = 2 and Q = 8
I.F. = `e ^(int^(P^dθ) = e^(int^(2^dθ) = e^(2θ)`
Solution of the given equation is
`r(I.F.) = int Q (I.F.) dθ + c`
`re^(2θ) = int 8 e^(2θ) dθ + c`
`re^(2θ) = 8 int e^(2θ) dθ + c`
`re ^(2θ) = 8e^(2θ)/2 + c`
`re ^(2θ) = 4e^(2θ) + c`
APPEARS IN
RELATED QUESTIONS
Verify that y = cx + 2c2 is a solution of the differential equation
xy (y + 1) dy = (x2 + 1) dx
(1 − x2) dy + xy dx = xy2 dx
(y + xy) dx + (x − xy2) dy = 0
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).
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
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
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
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.
xdx + 2y dx = 0
The solution of `dy/dx + x^2/y^2 = 0` 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
y2 dx + (xy + x2)dy = 0
`xy dy/dx = x^2 + 2y^2`
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
