Advertisements
Advertisements
Question
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Advertisements
Solution
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
RELATED QUESTIONS
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`
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.
Solve the differential equation `dy/dx -y =e^x`
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
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
If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\], then
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
The number of arbitrary constants in the particular solution of a differential equation of third order is
Which of the following differential equations has y = x as one of its particular solution?
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
(x + y − 1) dy = (x + y) dx
(1 + y + x2 y) dx + (x + x3) dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
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 number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
