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Miscellaneous Exercise 6(II) · Q123

Q.Obtain the differential equation by eliminating the arbitrary constants: (y−a)2=b(x+4)(y-a)^2=b(x+4)

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(y−a)2=b(x+4)(y-a)^2=b(x+4), two constants. Differentiate: 2(y−a)dydx=b2(y-a)\dfrac{dy}{dx}=b. Differentiate again: 2(dydx)2+2(y−a)d2ydx2=0⇒(y−a)=−(dy/dx)2d2y/dx22\left(\dfrac{dy}{dx}\right)^2+2(y-a)\dfrac{d^2y}{dx^2}=0\Rightarrow(y-a)=-\dfrac{(dy/dx)^2}{d^2y/dx^2}. Substituting into the first-derivative equation to get bb, then substituting both (y−a)(y-a) and bb back into the original relation and simplifying (as verif …

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