Advertisements
Advertisements
प्रश्न
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
उत्तर
We have,
\[\frac{dy}{dx} + \frac{y}{x} = x^2 \]
\[\Rightarrow \frac{dy}{dx} + \frac{1}{x}y = x^2 \]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get}\]
\[P = \frac{1}{x} \]
\[Q = x^2 \]
Now,
\[I . F . = e^{\int\frac{1}{x}dx} \]
\[ = e^{\log\left| x \right|} \]
\[ = x\]
So, the solution is given by
\[y \times I . F . = \int Q \times I . F . dx + C\]
\[ \Rightarrow yx = \int x^3 + C\]
\[ \Rightarrow xy = \frac{x^4}{4} + C\]
APPEARS IN
संबंधित प्रश्न
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
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.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 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
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
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 (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
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:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
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 ______.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
