A relation in dependent variables and independent variables satisfying a differential equation is called general solution of that differential equation.

True

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False

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Solution

The correct option is B False
We are making a statement about general solution of a differential equation is. Let us first look at what general solution of a differential equation is or how it is defined. A relation in x and y (or dependent variable and independent variable) with equal number of arbitrary constants as the order of DE is called general solution of a differential equation. The part which says the relation should have as many arbitrary variables as the order of the differential equation is missing in the statement given in the question. We will understand this better with an example.
For the differential equation $\frac{d y}{d x} = 2 x - - - (1), y = x^{2} + C$ is the general solution. $y = x^{2} + C$ satisfies the equation. It also has 1 arbitrary constant, same as the order of the differential equation. But the solution $y = x^{2}$ is not a general solution because it does not have any arbitrary constant.

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The first order linear differential equation, where y is independent and x is dependent variable, is:

\frac{d y}{d x} + \frac{d z}{d x} sin x = 0

y

z

\begin{matrix} Differential equation & General solution (A) 2 x y^{2} d x = e^{x} (d y - y d x) & (P) 4 x^{3} y + 6 e^{x} - 3 y^{3} = c y (B) y (2 x^{2} y + e^{x}) d x = (e^{x} + y^{3}) d y & (Q) x + {log}_{e} (x^{2} + y^{2}) = c (C) (x^{2} + y^{2} + 2 x) d x + 2 y d y = 0 & (R) x^{2} {log}_{e} (x^{2} y^{2}) + y^{3} = c x^{2} (D) (2 x^{2} y - 2 y^{4}) d x + (2 x^{3} + 3 x y^{3}) d y = 0 & (S) x^{2} y + e^{x} = c y \end{matrix}

c

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