Advertisements
Advertisements
प्रश्न
y dx – x dy + log x dx = 0
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
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 \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
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?
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`(x + y) dy/dx = 1`
The solution of `dy/ dx` = 1 is ______.
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
