87273 The general solution of differential equation
\(\frac{d y}{d x}+2 x y=2 e^{-x^{2}} \text { is }\)

87274 The integrating factor of the D.E.
\((x \log x) \frac{d y}{d x}+y=2 \log x\) is

87275 If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 e^{-4 x}(2 \sin 2 x+\cos 2 x)\) with, \(y(\pi / 4)=e^{-\pi}\), then \(y(\pi / 6)\) is equal to :

87276 Let the solution curve \(y=y(x)\) of the differential equation
\(\frac{d y}{d x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{\frac{3}{2}}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{6}\right)}}\right\}\)
pass through the origin. Then \(y(1)\) is equal to :

87273 The general solution of differential equation
\(\frac{d y}{d x}+2 x y=2 e^{-x^{2}} \text { is }\)

87274 The integrating factor of the D.E.
\((x \log x) \frac{d y}{d x}+y=2 \log x\) is

87275 If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 e^{-4 x}(2 \sin 2 x+\cos 2 x)\) with, \(y(\pi / 4)=e^{-\pi}\), then \(y(\pi / 6)\) is equal to :

87276 Let the solution curve \(y=y(x)\) of the differential equation
\(\frac{d y}{d x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{\frac{3}{2}}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{6}\right)}}\right\}\)
pass through the origin. Then \(y(1)\) is equal to :

87273 The general solution of differential equation
\(\frac{d y}{d x}+2 x y=2 e^{-x^{2}} \text { is }\)

87274 The integrating factor of the D.E.
\((x \log x) \frac{d y}{d x}+y=2 \log x\) is

87275 If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 e^{-4 x}(2 \sin 2 x+\cos 2 x)\) with, \(y(\pi / 4)=e^{-\pi}\), then \(y(\pi / 6)\) is equal to :

87276 Let the solution curve \(y=y(x)\) of the differential equation
\(\frac{d y}{d x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{\frac{3}{2}}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{6}\right)}}\right\}\)
pass through the origin. Then \(y(1)\) is equal to :

87273 The general solution of differential equation
\(\frac{d y}{d x}+2 x y=2 e^{-x^{2}} \text { is }\)

87274 The integrating factor of the D.E.
\((x \log x) \frac{d y}{d x}+y=2 \log x\) is

87275 If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 e^{-4 x}(2 \sin 2 x+\cos 2 x)\) with, \(y(\pi / 4)=e^{-\pi}\), then \(y(\pi / 6)\) is equal to :

87276 Let the solution curve \(y=y(x)\) of the differential equation
\(\frac{d y}{d x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{\frac{3}{2}}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{6}\right)}}\right\}\)
pass through the origin. Then \(y(1)\) is equal to :