# Solving Differential Equations with given initial conditions

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Differential Equations

5. Solve the differential equation with the given initial conditions:

6. Solve the differential equation with the given initial conditions:

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#### Solution Preview

Please show all work with explanations.

Differential Equations

5. Solve the differential equation with the given initial conditions:

6. Solve the differential equation with the given initial conditions:

SOLUTION:

Hi Dear,

The first step towards solving a differential equation is always to know what kind of differential equation it is, this is mainly because there are different methods to solving different kind of differential equations. I will here explain to you what kind of differential equations you have, and then progress to showing you how to solve them.

First of all you should notice that the two differential equations we need to solve are

and

(The other equations that follow them are just initial conditions)

Now, both of your equations are second-order because they have 2 as the highest order of the differentials (that is the ).

Next, they are both homogenous because they don't contain functions that are in t. For example, the equations; and are not homogenous because of the functions in t they contain (that is the 't' and '3t'). Your equations however do not contain functions in t, so they are homogenous (PLEASE NOT THAT WE ARE TALKING ...

#### Solution Summary

This solution is a concise illustration on how to solve differential equations with given initial conditions, specific illustrations were done using 2 problems of homogeneous second-order differential equations with constant coefficients.