Find the general solution of the given differential equation. drdθ+rsec⁡(θ)=cos⁡(θ) Give the largest over which the general solution is defined.

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Liek1993 · Accepted Answer

The equation is:  drdθ+rsec⁡θ=cos⁡θ  Integrating factor =e∫sec⁡θdθ  =eln⁡(sec⁡θ+tan⁡θ)  =sec⁡θ+tan⁡θ  The solution is given by:  r(sec⁡θ+tan⁡θ)=∫(sec⁡θ+tan⁡θ)cos⁡θdθ+c  =∫(1+sin⁡θ)dθ+c  =θ−cos⁡θ+c  Hence, the general solution is given as,  r(θ)=θsec⁡θ+tan⁡θ−cos⁡θsec⁡θ+tan⁡θ+csec⁡θ+tan⁡θ  The largest interval for which the solution is defined:  The solution does not exist for sec⁡θ+tan⁡θ=0  sec⁡θ is defined for all real numbers except integral multiple of π2  tan⁡θ is defined for all real numbers except integral multiple of π2  Thus, sec⁡θ+tan⁡θ is defined for all real numbers except integral multiple of π2  Hence, largest interval of existence is (−π2,π2)

Find the general solution of the given differential equation.\frac{dr}{d\theta}+r\sec(\theta)=\cos(\theta)

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