Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\Rightarrow \frac{dy}{dx} = 1 + x + y^2 \left( 1 + x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \left( 1 + x \right)\left( 1 + y^2 \right)\]
\[ \Rightarrow \frac{dy}{\left( 1 + y^2 \right)} = \left( 1 + x \right)dx\]
\[ \Rightarrow \int\frac{dy}{\left( 1 + y^2 \right)} = \int\left( 1 + x \right)dx\]
\[ \Rightarrow \tan^{- 1} y = x + \frac{x^2}{2} + C . . . . . \left( 1 \right)\]
\[\text{ Now, } \tan^{- 1} 0 = 0 + 0 + C ..........\left[\because y = 0, x = 0 \right]\]
\[ \Rightarrow C = 0\]
Substituting the value of C in (1), we get
\[ \tan^{- 1} y = x + \frac{x^2}{2}\]
\[ \Rightarrow y = \tan\left( x + \frac{x^2}{2} \right)\]
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 = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(ey + 1) cos x dx + ey sin x dy = 0
(1 − x2) dy + xy dx = xy2 dx
y (1 + ex) dy = (y + 1) ex dx
(y + xy) dx + (x − xy2) dy = 0
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
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 particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
x2 dy + y (x + y) dx = 0
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
y2 dx + (x2 − xy + y2) dy = 0
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
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` |
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` |
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
x2y dx – (x3 + y3) dy = 0
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
