Advertisements
Advertisements
प्रश्न
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).
Advertisements
उत्तर
According to the question,
\[\frac{dy}{dx} = x^2\]
\[\Rightarrow dy = x^2 dx\]
Integrating both sides with respect to x, we get
\[\int dy = \int x^2 dx\]
\[ \Rightarrow y = \frac{x^3}{3} + C\]
\[\text{ Since the curve passes through }\left( - 1, 1 \right), \text{ it satisfies the above equation .} \]
\[ \therefore 1 = \frac{- 1}{3} + C\]
\[ \Rightarrow C = 1 + \frac{1}{3}\]
\[ \Rightarrow C = \frac{4}{3}\]
Putting the value of C, we get
\[y = \frac{x^3}{3} + \frac{4}{3}\]
\[ \Rightarrow 3y = x^3 + 4\]
APPEARS IN
संबंधित प्रश्न
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Show that y = AeBx is a solution of the differential equation
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 = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
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.
(x + y) (dx − dy) = dx + dy
(x2 − y2) dx − 2xy dy = 0
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
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:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
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.
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 solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation satisfied by ax2 + by2 = 1 is
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
The solution of `dy/ dx` = 1 is ______.
`xy dy/dx = x^2 + 2y^2`
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Solve the differential equation `"dy"/"dx" + 2xy` = y
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
