Differentiate me

Calculus Level 4

d y d x = 2 x 2 + 17 y + 18 \large \dfrac{dy}{dx} = \dfrac{2x^2+17}{y+18}

If x x and y y satisfies the differential equation above, which of the following is a possible relationship between x x and y y ?

y = 17 3 ( 8 K + 2 x 3 + 200 x + 100 − 54 ) y=\frac{17}{3}\left(\sqrt{8} \sqrt{K+2x^3+200x+100} -54 \right) I don't know y = 1 3 ( 6 K + 2 x 3 + 9 x + 0 − 54 ) y=\frac{1}{3}\left(\sqrt{6} \sqrt{K+2x^3+9x+0} -54 \right) y = 1 3 ( 6 K + 2 x 3 + 51 x + 486 − 54 ) y=\frac{1}{3}\left(\sqrt{6} \sqrt{K+2x^3+51x+486} -54 \right)

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1 solution

Otto Bretscher
Apr 29, 2016

With the substitution z = y + 18 z=y+18 we have z d z = ( 2 x 2 + 17 ) d x zdz=(2x^2+17)dx and z 2 2 = 2 3 x 3 + 17 x + C \frac{z^2}{2}=\frac{2}{3}x^3+17x+C so y = z − 18 = ± 4 3 x 3 + 34 x + C − 18 y=z-18=\pm \sqrt{\frac{4}{3}x^3+34x+C}-18 , the third choice we are given. (The summand 486 is irrelevant since it is merely added to a constant.)

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