Advertisements
Advertisements
प्रश्न
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
उत्तर
We have,
\[\frac{dy}{dx} + y = 1\]
\[ \Rightarrow \frac{dy}{dx} = 1 - y\]
\[ \Rightarrow \frac{1}{\left( 1 - y \right)}dy = dx\]
Integrating both sides, we get
\[ - \int\frac{1}{\left( y - 1 \right)}dy = \int dx\]
\[ \Rightarrow \int\frac{1}{\left( y - 1 \right)}dy = - \int dx\]
\[ \Rightarrow \log \left| y - 1 \right| = - x + \log C\]
\[ \Rightarrow \log \left| y - 1 \right| - \log C = - x\]
\[ \Rightarrow \log \left| \frac{y - 1}{C} \right| = - x\]
\[ \Rightarrow \frac{y - 1}{C} = e^{- x} \]
\[ \Rightarrow y = 1 + C e^{- x}\]
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, 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 general solution of a differential equation of fourth 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.`
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 \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The number of arbitrary constants in the general solution of differential equation of fourth order is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
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`.
Solve: `2(y + 3) - xy "dy"/"dx"` = 0, given that y(1) = – 2.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` 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 differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
