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The differential equation of the family of curves, x^{2}=4b(y+b),b\; \epsilon \; R, is :
Option: 1 xy''=y'
Option: 2 x(y')^{2}=x+2yy'
Option: 3 x(y')^{2}=x-2yy'
Option: 4 x(y')^{2}=2yy'-x
 

Answers (1)

best_answer

 

 

Formation of Differential Equation and Solutions of a Differential Equation -


Differential equations are used to represent a family of curves. Suppose that a family of curves is represented using a set of n arbitrary constants, then the differential equation corresponding to that family is a relation between x, y and derivatives of y upto order n, not containing any of the n arbitrary constants.

Given a relation between the variables x, y and n arbitrary constants C1, C2, .... Cn, if we differentiate n times in succession with respect to x, we have altogether n + 1 equations between which the n arbitrary constants can be eliminated. The result is a differential equation of the nth order. 

Consider the equation of the ellipse

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(4)}\\\\\mathrm{differentiate\;Eq\;(4)\;w.r.t.\;'x'}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{2x}{a^2}+\frac{2y}{b^2}\cdot\left ( \frac{dy}{dx} \right )=0\;\;\;or\;\;\;\frac{x}{a^2}=-\frac{yy'}{b^2}\;\;\;\;\;\;\;\;\ldots(5)}\\\\\mathrm{differentiate\;above\;equation\;w.r.t.\;'x'}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{a^2}=-\frac{yy''+\left (y' \right )^2}{b^2} =0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(6)}\\\\\text{eliminating the constant 'a' and 'b' from (5) and (6), we get }\\\\\mathrm{yy'=xyy''+x(y')^2}

This is the differential equation of the family of the equation of ellipse (4).

From the above examples, it is clear that the order of the differential equation of the family of curve is equal to the number of arbitrary constant that presents in family of curve.

NOTE:

If arbitrary constants appear in addition, subtraction, multiplication or division, then we can club them to reduce into one new arbitrary constant. Hence, the differential equation corresponding to a family of curves will have order exactly same as number of essential arbitrary constants in the equation of the curve.

 

-

\\x^{2}=4 b(y+b)\\ \quad 2 x=4 b y' \\ \Rightarrow \quad b=\frac{x}{2 y'}

So the Differential equation is

x^{2}=\frac{2 x}{y'} \cdot y+\left(\frac{x}{y^{\prime}}\right)^{2} \\\Rightarrow x\left(\frac{d y}{d x}\right)^{2}=2 y \frac{d y}{d x}+x

Correct Option (2)

Posted by

vishal kumar

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