Advertisements
Advertisements
Question
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
Solution
Given differential equation is `x (dy)/(dx) - y = x^2.e^x`
⇒ `(dy)/(dx) - y/x` = xex, which is of the form
`(dy)/(dx) + Py` = Q
Here, P = `-1/x` and Q = xex
I.F. = `e^(intpdx)`
= `e^(int (-1)/x dx)`
= `e^(-logx)`
= `e^(log 1/x)`
The solution is given by y.IF. = `intQ xx I.F.dx + C`
`y. 1/x = intxe^x xx 1/x dx + C`
`y/x = inte^xdx + C`
`y/x = e^x + C`
`y/x = e^x` ...(i)
Given y = 0 when x = 1
From equation (i), we get
0 = 1.e1 + C.1
⇒ C = –e
y = xex – ex
RELATED QUESTIONS
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(x2 + 1) dy + (2y − 1) dx = 0
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
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:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), 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 ______.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
