Determine the particular solution to the differential equation , given . |
||
Identifier: The directions of the problem include the language “ particular solution ” and “ differential equation ”. |
||
Step 1: Identify your two variables. In this example the two variables are the y-variable and x-variable . |
|
|
Step 2: Separate your equation by variables on to either side of the equals. 1) Multiply both sides of the equation by to get the y-variables on the same side as . 2) Multiply both sides of the equation by to get the x-variables on the same side as . |
1) 2)
|
|
Step 3: Take the antiderivative of both sides of the equation. Both antiderivatives require you to run a u-substitution process. Keep in mind that you only need a single that works for the entire equation. |
|
|
|
|
|
|
|
|
|
||
Step 4: Solve the equation you found in Step 3 to get the dependent variable ( y -variable) alone on one side of the equals. |
|
|
Step 5 (If asked to find a particular solution): Determine the particular +C value for your specific situation.
In this example you were given the fact that . Which means that when , . 1) Plug in 1 for x and 2 for y in the general solution from Step 4, and evaluate any parts of the equation you can. Here you can simplify 2) Subtract one, , from both sides to isolate the part of the equation raised to the power. 3) Raise both sides to the power to cancel out the power. 4) Plug the C – value back into the general solution from Step 4. |
1)
2)
3)
4)
|
|
Final Result: The particular solution to the differential equation , given would be . |