Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Advertisements
उत्तर
`(x + a) dy/dx = – y + a`
∴ `dy/dx + y/((x+a)) = a / ((x+a))`
The given equation is of the form
`dy/ dx + py = Q`
where, `P = 1/((x+a)) and Q = a/((x+a))`
∴ I.F. = `e ^(int^(pdx) = e ^(int^(1/(x+a))^dx)`
= `e^(log^ |x+a|) = (x+a)`
∴ Solution of the given equation is
`y ( I.F.) = int Q (I.F.) dx + c `
∴ `y(x + a) = int a/((x+a)) (x+a) dx + c`
∴ `y(x + a) = a int 1 dx + c`
∴ y (x + a) = ax + c
APPEARS IN
संबंधित प्रश्न
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
x cos y dy = (xex log x + ex) dx
tan y dx + sec2 y tan x dy = 0
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
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).
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
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.
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
The solution of `dy/ dx` = 1 is ______.
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation xdx + 2ydy = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
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 ______.
Why is the equation \[x\frac{dy}{dx} + y = 0\] classified as a differential equation?
In mathematics and science, what primary purpose do differential equations serve?
