Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[x\frac{dy}{dx} = x + y\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x + y}{x}\]
This is a homogeneous differential equation .
\[\text{ Putting }y = vx\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx},\text{ we get }\]
\[v + x\frac{dv}{dx} = \frac{x + vx}{x}\]
\[ \Rightarrow v + x\frac{dv}{dx} = 1 + v\]
\[ \Rightarrow x\frac{dv}{dx} = 1 + v - v\]
\[ \Rightarrow x\frac{dv}{dx} = 1\]
\[ \Rightarrow dv = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int dv = \int\frac{1}{x}dx\]
\[ \Rightarrow v = \log \left| x \right| + C\]
\[\text{Putting }v = \frac{y}{x},\text{ we get}\]
\[ \Rightarrow \frac{y}{x} = \log \left| x \right| + C\]
\[ \Rightarrow y = x\log \left| x \right| + Cx \]
\[\text{ Hence, }y = x\log \left| x \right| + Cx\text{ is the required solution }.\]
APPEARS IN
संबंधित प्रश्न
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
(1 − x2) dy + xy dx = xy2 dx
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
A population grows at the rate of 5% per year. How long does it take for the population to double?
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
Solve the following differential equation.
dr + (2r)dθ= 8dθ
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
The solution of `dy/ dx` = 1 is ______.
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
Solve the differential equation:
`e^(dy/dx) = x`
Solve:
(x + y) dy = a2 dx
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`
∴ (1) becomes `"dv"/("d"x) - 1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
