Advertisements
Advertisements
प्रश्न
y dx – x dy + log x dx = 0
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
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
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 differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
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
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
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.
Solve the following differential equation.
x2y dx − (x3 + y3 ) dy = 0
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation:
`e^(dy/dx) = x`
Solve
`dy/dx + 2/ x y = x^2`
Solve the following differential equation y2dx + (xy + x2) dy = 0
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation
`x + y dy/dx` = x2 + y2
