Advertisements
Advertisements
प्रश्न
Verify that y = cx + 2c2 is a solution of the differential equation
उत्तर
We have,
\[y = cx + 2 c^2..............(1)\]
Differentiating both sides of (1) with respect to x, we get
Now,
\[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y\]
\[ = 2 c^2 + cx - cx - 2 c^2 = 0 ...........\left[\text{Using }\left( 1 \right)\text{ and }\left( 2 \right) \right]\]
\[ \Rightarrow 2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0\]
Hence, the given function is the solution to the given differential equation.
APPEARS IN
संबंधित प्रश्न
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
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.
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
C' (x) = 2 + 0.15 x ; C(0) = 100
(1 + x2) dy = xy dx
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
x2 dy + y (x + y) dx = 0
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The differential equation satisfied by ax2 + by2 = 1 is
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`dy/dx + y` = 3
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
