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Question Number 175406 by princeDera last updated on 29/Aug/22

$$\frac{d^2\mathit{y}}{{\mathit{d}\mathit{x}}^2}-\tan \left(\mathit{x}\right)\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=0$$

Answered by Ar Brandon last updated on 29/Aug/22

$$y^{″}-\tan x\cdot y^{\prime }=0$$$$\Rightarrow \frac{y^{″}}{y^{\prime }}=\tan x$$$$\Rightarrow \ln \left(y^{\prime }\right)=-\ln \left(\cos x\right)+C$$$$\Rightarrow \frac{d\mathrm{y}}{dx}=\frac{1}{\cos x}{e}^C=\frac{k}{\cos x}$$$$\Rightarrow \mathrm{y}=\int k\sec xdx=k\ln \mid \sec x+\tan x)+k_2$$

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