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प्रश्न
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
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उत्तर
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called particular solution.
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 = cx + 2c2 is a solution of the differential equation
(1 + x2) dy = xy dx
(y + xy) dx + (x − xy2) dy = 0
(x + y) (dx − dy) = dx + dy
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
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.
`dy/dx = x^2 y + y`
The solution of `dy/dx + x^2/y^2 = 0` is ______
Solve the differential equation:
`e^(dy/dx) = x`
x2y dx – (x3 + y3) dy = 0
`xy dy/dx = x^2 + 2y^2`
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
