Advertisements
Advertisements
प्रश्न
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} = y \tan x\]
\[ \Rightarrow \frac{1}{y}dy = \tan x dx\]
Integrating both sides, we get
\[\int\frac{1}{y}dy = \int\tan x dx\]
\[ \Rightarrow \log y = \log \left| \sec x \right| + C . . . . . . . \left( 1 \right)\]
Now,
When `x = 0, y = 1`
\[ \therefore \log 1 = \log 1 + C\]
\[ \Rightarrow C = 0\]
Putting the value of `C` in (1), we get
\[\log y = \log \left| \sec x \right|\]
\[ \Rightarrow y = \sec x\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
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.
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
cos (x + y) dy = dx
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
(x3 − 2y3) dx + 3x2 y dy = 0
x2 dy + (x2 − xy + y2) dx = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
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 ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
