The function y(t) satisfies the differential equation dy/dt=y4-3y3-18y2 List the

Question

The function y(t) satisfies the differential equation dy/dt=y4-3y3-18y2 List the (distinct) constant solutions (y = c) to the differential equation in ascending order. (If there are fewer than four solutions, leave the latter blanks empty) For what values of y (in interval notation) is y increasing? Use the strings "plus_infinity" or "minus_infmity" if appropriate, and if there is only one interval, leave the second one blank. Finally, list your intervals so that the first interval is to the left of the second (on the real line). Interval 1: ( , ) Interval 2:( , )

Explanation / Answer

DY/DT = Y^4-3Y^3-18Y^2=[Y^2][Y^2-3Y-18]=[Y^2][Y-6][Y+3]........... SO THE CONSTANT SOLUTIONS ARE ....Y=-3 , 0 , 0 , 6........... THE CRITICAL POINTS ARE Y=-3,0,6 SO - INFINITY TO -3 .......Y' IS POSITIVE ....SO FUNCTION WILL INCREASE ... -3 TO 0 ............Y' IS NEGATIVE ......SO FUNCTION WILL DECREASE ... 0 TO 6 .....Y' IS NEGATIVE .....SO FUNCTION WILL DECREASE 6 TO + INFINITY .....Y' IS POSITIVE ...SO FUNCTION WILL INCREASE

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