Advertisements
Advertisements
Question
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Advertisements
Solution
`("d"y)/("d"x)` = x2y + y
∴ `("d"y)/("d"x)` = y(x2 + 1)
∴ `("d"y)/y` = (x2 + 1) dx
Integrating on both sides, we get
`int ("d"y)/y = int(x^2 + 1) "d"x`
∴ log |y| = `x^3/3 + x + "c"`
APPEARS IN
RELATED QUESTIONS
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
(sin x + cos x) dy + (cos x − sin x) dx = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
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 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).
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
3x2 dy = (3xy + y2) dx
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
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?
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
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.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
Solve
`dy/dx + 2/ x y = x^2`
y dx – x dy + log x dx = 0
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Solve: ydx – xdy = x2ydx.
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
