### Home > CALC > Chapter 7 > Lesson 7.3.5 > Problem7-162

7-162.

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

1. Find 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 $y$.

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

3. Find 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, $y =\operatorname{ln}x$, is $x > 0$.