CPM Homework Help : APCALC Problem 7-157

Solve for $y$ if $\frac { d y } { d x }= 0.5x^2$ .

Then use the slope field for $\frac { d y } { d x }$ at right to help graph a family of solution functions (place your paper over the slope field and use the tangents as guides).

Hint 1:

By tracing a series of slopes, you should predict that the original function is cubic.

Hint 2:

Separate the variables so that $x$ -values are on one side of the equal sign and $y$ -values are on the other side.

Step 1:

$2dy = x^2dx$

Hint 3:

Integrate both sides of the equation.

Step 2:

$\int 2dy=\int x^{2}dx$

$2y+C_{1}=\frac{x^{3}}{3}+C_{2}$

Hint 4:

Solve for $y$ . Did you get a cubic function, as predicted?
Note: $C_1 + C_2 = C_3$ , or just $C$ .