Explanation The given equation is y 2 − x = C . Differentiate both sides of the equation y 2 − x = C with respect to x . d d x ( y 2 − x ) = d d x C d d x y 2 − d d x x = 0 2 y y ′ − 1 = 0 2 y y ′ = 1 The given differential equation obtained. Thus, any function y defined by the y 2 − x = C is the solution of the differential equation 2 y y ′ = 1 . The initial condition is y ( 0 ) = 3 . So, substitute x = 0 and y = 3 in y 2 − x = C as follows. y 2 − x = C 3 2 − 0 = C 9 = C Thus, the particular solution satisfying y ( 0 ) = 3 is a solution of the equation y 2 − x = 9

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Chapter 9.1, Problem 4MP

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To show: The function y satisfies the differential equation 2yy′=1 if y is defined implicitly by the equation y2−x=C and find the explicit solution that satisfies the initial condition y(0)=3.

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The function y satisfies the differential equation 2 y y ′ = 1 if y is defined implicitly by the equation y 2 − x = C and find the explicit solution that satisfies the initial condition y ( 0 ) = 3 .

Solution Summary: The author explains that if y is defined implicitly by the equation y2-x=C and finds the explicit solution, the function satisfies the initial condition.

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To show: The function y satisfies the differential equation 2yy′=1 if y is defined implicitly by the equation y2−x=C and find the explicit solution that satisfies the initial condition y(0)=3.

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