Can someone show me how to find the general solution of the differential equations? really need to know how to do it for the upcoming exam

The first equation is linear:



Divide through by to get



and notice that the left hand side can be consolidated as a derivative of a product. After doing so, you can integrate both sides and solve for .





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The second equation is also linear:



Multiply both sides by to get



and recall that , so we can write





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Yet another linear ODE:



Divide through by , giving








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In case the steps where we multiply or divide through by a certain factor weren't clear enough, those steps follow from the procedure for finding an integrating factor. We start with the linear equation



then rewrite it as



The integrating factor is a function such that



which requires that



This is a separable ODE, so solving for we have





and so on.