Question

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52.

31.$\frac{dy}{dx}=-6x^2,y\left(1\right)=3$

Short answer

The answer is$y\left(x\right)=-2x^3+5$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=-6x^2,y\left(1\right)=3$

Step 2. Solve using antidifferentiation and/or variable separable method.

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int dy==\int -6x^{2}dx \\ y=-6·\frac{1}{3}{x}^3+C \\ y=-2x^3+C \end{gathered}$$

Now put $y\left(1\right)=3$,

$$\begin{gathered} 3=-2{\left(1\right)}^3+C \\ 3+2=C \\ 5=C \end{gathered}$$

So, the answer is$y\left(x\right)=-2x^3+5$