Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}\]
\[ \Rightarrow dy = \left( x^5 + x^2 - \frac{2}{x} \right)dx\]
Integrating both sides, we get
\[ \Rightarrow \int dy = \int\left( x^5 + x^2 - \frac{2}{x} \right)dx\]
\[ \Rightarrow y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + C\]
\[\text{Clearly, } y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + \text{ C is defined for all }x \in R \text{ except }x = 0 . \]
\[\text{ Hence, }y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + \text{ C, where } x \in R - \left\{ 0 \right\}, \text{ is the solution to the given differential equation.}\]
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
(x2 − y2) dx − 2xy dy = 0
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
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.
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
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?
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
Define a differential equation.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve: `("d"y)/("d"x) + 2/xy` = x2
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
Solve the differential equation
`x + y dy/dx` = x2 + y2
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
