### Home > APCALC > Chapter 7 > Lesson 7.4.1 > Problem7-185

7-185.

No calculator! Suppose $\frac { d y } { d x } = \frac { 3 x ^ { 2 } } { e ^ { y } }$.

1. Write the particular solution of this differential equation containing the point $(0, 3)$.

Cross multiply in order to separate the variables.

Integrate both sides.

Solve for $C$ by evaluating the given point.

2. If you have not already done so, solve your equation for $y$. Confirm that your solution is correct by substituting into the differential equation.

$y = \ln(x^3 + e^3)$

3. State the domain and range of your equation.

The domain is the set of values $x$ can be so that $y$ is still defined.
The range is the set of $y$ values resulting from inputting all members of the domain in the function.

The domain of the parent graph, $y = \ln(x)$, is $x > 0$.