To solve this differential equation use separation of variables. This means move all terms containingto one side of the equation and all terms containingto the other side.

First, multiply each side by.

Now divide byon both sides.

Next, divide byon both sides.

From here take the integral of both sides. Remember rules for logarithmic functions as they will be used in this problem.

年代o this is a separable differential equation. The first step is to move all of the x terms (including dx) to one side, and all of the y terms (including dy) to the other side.

年代o the differential equation we are given is:

Which rearranged looks like:

At this point, in order to solve for y, we need to take the anti-derivative of both sides:

Which equals:

And since this an anti-derivative with no bounds, we need to include the general constant C

年代o, solving for y, we raise e to the power of both sides:

which, simplified gives us our answer:

The simplest way to solve a separable differential equation is to rewriteasand, by an abuse of notation, to "multiply both sides by dt". This yields

.

Next, we get all the y terms with dy and all the t terms with dt and integrate. Thus,

Combining the constants of integration and exponentiating, we have

The plus minus and thecan be combined into another arbitrary constant, yielding.

Plugging in our initial condition, we have

and

First the differential equation can be separated to:

And then integrated simply to:

Using exponential rules, we note thatbecomes. Meaning that the

differential equation is equivalent to:

which by separation of variables is:

The differential equationcannot be written asand is therefore not separable.

To solve this differential equation use separation of variables. This means move all terms containingto one side of the equation and all terms containingto the other side.

First, multiply each side by.

Now divide byon both sides.

Next, divide byon both sides.

From here take the integral of both sides. Remember rules for logarithmic functions as they will be used in this problem.

This is a separable differential equation. The simplest way to solve this is to first rewriteasand then by an abuse of notation to "multiply both sides by dt." This yields. Then group all the y terms with dy and integrate, getting us to. Solving for y, we have. Plugging in our condition, we find. Raising both sides to the power of -1/3, we see. Thus, our final solution is

This is a separable ODE, so rearranging

Integrating

Plugging in the initial condition and solving gives us

年代olving forgives us

First, divide byon both sides of the equation.

Identify the factorterm.

Integrate the factor.

年代ubstitute this value back in and integrate the equation.

Now divide byto get the general solution.

The transient term means a term that when the values get larger the term itself gets smaller. Therefore the transient term for this function is.