Solution to this differential equation that does not diverge at&nbsp;x=0.tanx&nbsp;dydx+y=extanxBy using the integrating factor&nbsp;μ(x)=cos⁡x, I solved it as an equation in full differentials and got the solutiony(x)=Acsc⁡x+12ex(1−cot⁡x)However, the question I am solving asks for a solution that does not diverge at&nbsp;x=0, which this solution clearly does because of the cotx. How can I get a solution that converges?

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Solution to this differential equation that does not diverge at&amp;nbsp;x=0.tanx&amp;nbsp;dydx+y=extanxBy using the integrating factor&amp;nbsp;μ(x)=cos⁡x, I solved it as an equation in full differentials and got the solutiony(x)=Acsc⁡x+12ex(1−cot⁡x)However, the question I am solving asks for a solution that does not diverge at&amp;nbsp;x=0, which this solution clearly does because of the cotx. How can I get a solution that converges?

Jonas Dickerson · Accepted Answer

Step 1Notice thatAcsc(x)+12ex(1-cot(x))=Asin(x)+ex2(sin(x)-cos(x)sin(x))=A+ex(sin(x)-cos(x))2sin(x).Step 2Notice that the divergence at 0 occurs because&amp;nbsp;sin⁡(0)=0. Consider the numerator,A+ex(sin(x)-cos(x))2.At 0, this numerator is equal to&amp;nbsp;A−12. This leaves you with the only option&amp;nbsp;A=12, to ensure the singularity at 0 becomes removable.

Rey Mcmillan · Accepted Answer

At x=0, we want A csc⁡ x=12excot⁡x. By representing csc x and cot x in their complex exponential form, and substituting x=0, we get A=12.

Solution to this differential equation that does not diverge at x=0. \tan x \; \frac{\t

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