Advertisements
Advertisements
Question
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Advertisements
Solution
`(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
RELATED QUESTIONS
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(sin x + cos x) dy + (cos x − sin x) dx = 0
(y2 + 1) dx − (x2 + 1) dy = 0
x2 dy + y (x + y) dx = 0
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
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.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
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
y2 dx + (xy + x2)dy = 0
`dy/dx = log x`
y dx – x dy + log x dx = 0
Solve the differential equation xdx + 2ydy = 0
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`
∴ (1) becomes `"dv"/("d"x) - 1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Solve the differential equation `"dy"/"dx" + 2xy` = y
A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is
