Advertisements
Advertisements
प्रश्न
Find the general solution of `"dy"/"dx" + "a"y` = emx
Advertisements
उत्तर
Given equation is `"dy"/"dx" + "a"y` = emx
Here, P = a and Q = emx
∴ I.F. = `"e"^(int Pdx)`
= `"e"^(int a .dx)`
= eax.
Solution of equation is `y xx "I"."F" = int "Q" "I"."F" "d"x + "c"`
⇒ `y."e"^("a"x) = int "e"^"mx" . "e"^("a"x) "d"x + "c"`
⇒ `y . "e"^("a"x) = int "e"^(("m" + "a")x) "d"x + "c"`
⇒ `y . "e"^("a"x) = "e"^(("m" + "a")x)/(("m" + "a")) + "c"`
⇒ y = `"e"^(("m" + "a")x)/(("m" + "a")) . "e"^(-"a"x) + "c"."e"^(-"a"x)`
∴ y = `"e"^("m"x)/(("m" + "a")) + "c" . "e"^(-"a"x)`
APPEARS IN
संबंधित प्रश्न
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
If y = etan x+ (log x)tan x then find dy/dx
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
cos (x + y) dy = dx
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
(1 + y + x2 y) dx + (x + x3) dy = 0
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
