Function​ y(x) Satisfying d ^3y/dx^3=12,

Question

Find the function​ y(x) satisfying d³y/dx³=12, y′′(0)=16, y′(0)=7, and y(0)=2.

y(x)=

Answer 1

If the third derivative is 12, then can you see that the second derivative must be 12x + C? Since y''(0) = 16 and y''(x) = 12x + C then 16 = 12(0) + C or C = 16. That makes y''(x) = 12x + 16.

If the second derivative is 12x + 16, can you see that the first derivative must be 6x² + 16x + C? Since y'(0) = 7 and y'(x) = 6x² + 16x + C then 7 = 6(0)² + 16(0) + C so C = 7. That makes y'(x) = 6x² + 16x + 7

If the first derivative is 6x² + 16x + 7, then can you see that the function y must be 2x³ + 8x² + 7x + C? Since y(0) = 2 and y(x) = 2x³ + 8x² + 7x + C then 2 = 2(0)³ + 8(0)² + 7(0) + C so C = 2.

So y(x) = 2x³ + 8x² + 7x + 2