Advertisements
Advertisements
प्रश्न
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
पर्याय
e−x
e−y
\[\frac{1}{x}\]
x
Advertisements
उत्तर
\[ \Rightarrow \frac{dy}{dx} - \frac{1}{x}y = 2x\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = - \frac{1}{x} \]
\[Q = 2x\]
Now,
\[I . F . = e^{- \int\frac{1}{x}dy} \]
\[ = e^{- \log\left| x \right|} \]
\[ = e^{log\left| \frac{1}{x} \right|} \]
\[ = \frac{1}{x}\]
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\] are rectangular hyperbola.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
The solution of `dy/ dx` = 1 is ______.
The solution of `dy/dx + x^2/y^2 = 0` is ______
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
Solve
`dy/dx + 2/ x y = x^2`
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Why is the equation \[x\frac{dy}{dx} + y = 0\] classified as a differential equation?
In mathematics and science, what primary purpose do differential equations serve?
