Advertisements
Advertisements
Question
y dx – x dy + log x dx = 0
Advertisements
Solution
y dx – x dy + log x dx = 0
y dx – x dy = - log x dx
Dividing throughout by dx, we get
`y-x dy/dx = – log x `
∴ `-xdy/dx + y = - log x`
∴ `dy/dx - 1/(x y) = logx/x`
The given equation is of the form
`dy/dx + py = Q`
where, `P = -1/x and Q = logx/x`
∴ I.F. = `e ^(int^(pdx) = e^(int^(-1/xdx) e ^-logx`
= `e^(logx ^-1) = x ^-1 = 1/x`
∴ Solution of the given equation is
`y(I.F.) =int Q (I.F.) dx + c`
∴ `y/x = int logx/x xx1/xdx+c`
In R. H. S., put log x = t …(i)
∴ x = et
Differentiating (i) w.r.t. x, we get
`1/xdx = dt`
∴ `y/x = int t/e^t dt +c`
∴ `y/x = int te^t dt +c`
= `t int e^-t dt - int (d/dt(t)xxint e^-t dt) dt +c `
= `-te^-t - int (-e^-t) dt +c`
= `-te^-t + int e^-t dt +c`
= – te–t – e –t + c
= `(-t-t)/e^t + c`
= `(- logx -1)/x +c`
∴ y = cx – (1 + log x)
APPEARS IN
RELATED QUESTIONS
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
(1 − x2) dy + xy dx = xy2 dx
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:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
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.
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
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.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
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
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
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
Solve
`dy/dx + 2/ x y = x^2`
y2 dx + (xy + x2)dy = 0
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
