y^{''}-2/r y^'+2/(r^2)y=0,y(1)=1,y(2)=0

解

y^{^{''}} - \frac{2}{r} y^{^{'}} + \frac{2}{r ^{2}} y = 0, y (1) = 1, y (2) = 0

y = e^{\frac{t}{r}} (\frac{sin ( \frac{2}{r} ) cos ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) )} - \frac{cos ( \frac{2}{r} ) sin ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) + sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) )})

解答ステップ

y^{''} (t) - \frac{2}{r} y^{'} (t) + \frac{2}{r ^{2}} y = 0

線形 ODE を解く:

y = e^{\frac{t}{r}} (\frac{sin ( \frac{2}{r} ) cos ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) )} - \frac{cos ( \frac{2}{r} ) sin ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) + sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) )})

y = e^{\frac{t}{r}} (\frac{sin ( \frac{2}{r} ) cos ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) )} - \frac{cos ( \frac{2}{r} ) sin ( \frac{t}{r} )}{e ^{\frac{1}{r}} ( - cos ( \frac{2}{r} ) sin ( \frac{1}{r} ) + sin ( \frac{2}{r} ) cos ( \frac{1}{r} ) )})