a) The solution of this <em>ordinary</em> differential equation is ![y =\sqrt[3]{-\frac{2}{\frac{3\cdot t}{8}-\frac{\sin 2t}{4}+\frac{\sin 4t}{32}-2   } }](https://tex.z-dn.net/?f=y%20%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7B2%7D%7B%5Cfrac%7B3%5Ccdot%20t%7D%7B8%7D-%5Cfrac%7B%5Csin%202t%7D%7B4%7D%2B%5Cfrac%7B%5Csin%204t%7D%7B32%7D-2%20%20%20%7D%20%7D) .
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b) The integrating factor for the <em>ordinary</em> differential equation is  .
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The <em>particular</em> solution of the <em>ordinary</em> differential equation is  .
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<h3>
How to solve ordinary differential equations</h3>
a) In this case we need to separate each variable ( ) in each side of the identity:
) in each side of the identity:
 (1)
 (1)

Where  is the integration constant.
 is the integration constant.
By table of integrals we find the solution for each integral:

If we know that  and
 and  <em>, </em>then the integration constant is
<em>, </em>then the integration constant is  .
.
The solution of this <em>ordinary</em> differential equation is ![y =\sqrt[3]{-\frac{2}{\frac{3\cdot t}{8}-\frac{\sin 2t}{4}+\frac{\sin 4t}{32}-2   } }](https://tex.z-dn.net/?f=y%20%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7B2%7D%7B%5Cfrac%7B3%5Ccdot%20t%7D%7B8%7D-%5Cfrac%7B%5Csin%202t%7D%7B4%7D%2B%5Cfrac%7B%5Csin%204t%7D%7B32%7D-2%20%20%20%7D%20%7D) .
. 
b) In this case we need to solve a first order ordinary differential equation of the following form:
 (2)
 (2)
Where:
 - Integrating factor - Integrating factor
 - Particular function - Particular function
Hence, the ordinary differential equation is equivalent to this form:
 (3)
 (3)
The integrating factor for the <em>ordinary</em> differential equation is  .
. 
The solution for (2) is presented below:
 (4)
 (4)
Where  is the integration constant.
 is the integration constant.
If we know that  and
 and  , then the solution of the ordinary differential equation is:
, then the solution of the ordinary differential equation is:



If we know that  and
 and  , then the particular solution is:
, then the particular solution is:

The <em>particular</em> solution of the <em>ordinary</em> differential equation is  .
. 
To learn more on ordinary differential equations, we kindly invite to check this verified question: brainly.com/question/25731911