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Question Number 153542 by EDWIN88 last updated on 08/Sep/21

$$y^{‴}+y^{\prime }=\sec x$$

Answered by puissant last updated on 08/Sep/21

$$y^{‴}+y^{\prime }=secx$$$$\Rightarrow y^{″}+y=ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)+C_1$$$$\Rightarrow (D^2+D)y=ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)+C_1$$$$\Rightarrow (D+1)Dy=ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)+C_1$$$$\Rightarrow (D+1)y=\int ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)dx+C_{1}x+C_2$$$$\Rightarrow e^x(D+1)y=e^x\int ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)dx+C_1{xe}^x+C_2{e}^x$$$$\Rightarrow D\left(e^{x}y\right)=e^x\int ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)dx+C_1{xe}^x+C_2{e}^x$$$$\Rightarrow e^{x}y=\int e^x\int ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)dxdx+\frac{C_1{x}^2{e}^x}{2}+C_2{xe}^x+C_3$$$$\Rightarrow y=e^{-x}\int e^x\int ln\left(tan(\frac{x}{2}+\frac{\pi }{2})\right)dxdx+\frac{C_1{x}^2}{2}+C_{2}x+C_3{e}^{-x}$$$$..................$$

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