Advertisements
Advertisements
प्रश्न
Solve
`dy/dx + 2/ x y = x^2`
Advertisements
उत्तर
`dy/dx + 2/ x y = x^2`
The given equation is of the form
`dy/dx + py = Q`
`where, P = 2/x and Q = x^2`
∴ I.F. =`e^(int^(pdx) = e^(2int^(1/xdx) e = ^(2logx) = e^(logx^2) = x^2`
∴ Solution of the given equation is
`y(I.F.) = int Q(I.F.) dx + c_1`
`y(x^2) = int x^2 xx x^2 dx + c_1`
∴ `x ^2 y = x^4 intdx + c_1`
∴ `x^2 y = x^5/5 + c_1`
∴ 5x2 y = x5 + c …[c = 5c1]
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
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 y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
(x + y) (dx − dy) = dx + dy
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
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:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
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.
The solution of the differential equation y1 y3 = y22 is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 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` |
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation
`x + y dy/dx` = x2 + y2
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
