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Question Number 188889 by mnjuly1970 last updated on 08/Mar/23

$$$$$$If,y=\frac{Arcsin\left(\sqrt{x}\right)}{\sqrt{x(1-x)}}\Rightarrow$$$$y^{\prime }.p\left(x\right)+y.q\left(x\right)=1$$$$$$$$find,{\int }_0^{1}p\left(x\right).q\left(x\right)dx=?$$$$p,qaretwopllynomils...$$$$$$

Answered by qaz last updated on 08/Mar/23

$$y=\frac{\arcsin \sqrt{x}}{\sqrt{x(1-x)}}\Rightarrow y^{\prime }=\frac{\frac{1}{2}-\frac{1-2x}{2\sqrt{x(1-x)}}\arcsin \sqrt{x}}{x(1-x)}$$$$=\frac{1-(1-2x)y}{2x(1-x)}$$$$\Rightarrow 2x(1-x)y^{\prime }+(1-2x)y=1$$$$\Rightarrow p\left(x\right)=2x(1-x)q\left(x\right)=1-2x$$$$\Rightarrow {\int }_0^{1}p\left(x\right)q\left(x\right)dx={\int }_0^{1}2x(1-x)(1-2x)dx=0$$

Commented by mnjuly1970 last updated on 08/Mar/23

$$merceysir$$

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