Advertisements
Advertisements
Question
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Options
y = `"e"^x (x - 1)`
y = xex
y = `x"e"^-x + 1`
y = xe–x
Advertisements
Solution
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is y = xe–x .
Explanation:
The given differential equation is `("d"y)/("d"x) + y = "e"^-x`
Since, it is a linear differential equation then P = 1 and Q = `"e"^-x`
Integrating factor I.F. = `"e"^(int Pdx)`
= `'e"^(int 1. "d"x)`
= ex
∴ Solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y xx "e"^x = int"e"^-x xx "e"^x"d"x + "c"`
⇒ `y xx "e"^x = int "e"^0 "d"x + "c"`
⇒ `y xx "e"^x = int 1."d"x + "c"`
⇒ `y xx "e"^x = x + "c"`
Put y = 0 and x = 0
∴ 0 = 0 + c
∴ c = 0
∴ Equation is `y xx "e"^x` = x
So y = `x"e"^-x`.
APPEARS IN
RELATED QUESTIONS
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
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`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
x2 dy + (x2 − xy + y2) dx = 0
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
