Advertisements
Advertisements
प्रश्न
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
विकल्प
y = ex(x – 1)
y = xe–x
y = xe–x + 1
y = (x + 1)e–x
Advertisements
उत्तर
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
∴ P = 1 and Q = e–x
∴ I.F = `"e"^(int 1."d"x)` = ex
So, the solution is `y xx "I"."F". = int "Q". "I"."F". "d"x + "c"`
⇒ `y . "e"^x = int"e"^-x . "e"^x "d"x + "c"`
⇒ `y . "e"^x = int 1."d"x + "c"`
⇒ `y . "e"^x + "c"`
Put x = 0, y = 0
We have 0 = 0 + c
∴ c = 0
So, the solution is `y "e"^x` = x
⇒ y = `x . "e"^-x`
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
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`
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 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
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] 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 solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
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 \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
\[\frac{dy}{dx} + 1 = e^{x + y}\]
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + 2y = \sin 3x\]
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
General solution of `("d"y)/("d"x) + y` = sinx 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 ______.
