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Miscellaneous Exercise 6(II) · Q121

Q.Obtain the differential equation by eliminating the arbitrary constants: y2=a(b−x)(b+x)y^2=a(b-x)(b+x)

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y2=a(b−x)(b+x)=a(b2−x2)y^2=a(b-x)(b+x)=a(b^2-x^2), two constants. Differentiate: 2ydydx=−2ax⇒a=−y dy/dxx2y\dfrac{dy}{dx}=-2ax\Rightarrow a=-\dfrac{y\,dy/dx}{x}. Differentiate again: (dydx)2+yd2ydx2=−a\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}=-a. Substituting aa from the first: (dydx)2+yd2ydx2=y dy/dxx\left(\dfrac{dy}{dx}\right)^2+y\dfrac{d^2y}{dx^2}=\dfrac{y\,dy/dx}{x}, i.e. $xy\dfrac{d^2y}{d …

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