Obtain the differential equation by eliminating the arbitrary constants: Ax²+By²=1

Exercise 6.2 · Q13

Ax^2+By^2=1

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✓ Free question

$Ax^2+By^2=1$ has two arbitrary constants, so differentiate twice. First: $2Ax+2By\dfrac{dy}{dx}=0$ , i.e. $Ax+By\dfrac{dy}{dx}=0$ . Differentiate again: $A+B\left[\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}\right]=0$ , so $A=-B\left[\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}\right]$ . From the first equation, $A=-\dfrac{By\,dy/dx}{x}$ . Equating the two expressions for $A$ and cancelling $B$ : $\dfrac{y\,dy/dx}{x}=\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}$ , i.e. $xy\dfrac{d^2y}{dx^2}+x\left(\dfrac{dy}{dx}\right)^2-y\dfrac{dy}{dx}=0$ .

✓Final answer

xy\dfrac{d^2y}{dx^2}+x\left(\dfrac{dy}{dx}\right)^2-y\dfrac{dy}{dx}=0

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Q.Obtain the differential equation by eliminating the arbitrary constants: $Ax^2+By^2=1$

Concept understanding — Formation of a Differential Equation whose General Solution is Given

Q.Obtain the differential equation by eliminating the arbitrary constants: Ax2+By2=1Ax^2+By^2=1Ax2+By2=1

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Q.Obtain the differential equation by eliminating the arbitrary constants: $Ax^2+By^2=1$

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