x^2y^{''}+xy^'-y= 1/(x+1)

Lösung

x^{2} y^{^{''}} + x y^{^{'}} - y = \frac{1}{x + 1}

y = c_{1} x + \frac{c _{2}}{x} - \frac{1}{2} x ln (x) - \frac{1}{2} + \frac{1}{2} x ln (x + 1) - \frac{ln ( x + 1 )}{2 x}

Schritte zur Lösung

x^{2} y^{''} (x) + x y^{'} (x) - y = \frac{1}{x + 1}

Solve linear ODE:

y = c_{1} x + \frac{c _{2}}{x} - \frac{1}{2} x ln (x) - \frac{1}{2} + \frac{1}{2} x ln (x + 1) - \frac{ln ( x + 1 )}{2 x}

y = c_{1} x + \frac{c _{2}}{x} - \frac{1}{2} x ln (x) - \frac{1}{2} + \frac{1}{2} x ln (x + 1) - \frac{ln ( x + 1 )}{2 x}

x \to 1 - lim (\frac{x}{ln ( x )})

\int \frac{( x + 3 )}{2 x ^{3} - 8 x} d x

\frac{\partial}{\partial y} (x e^{z})

\frac{d}{d x} (r^{2} - x^{2})

\frac{\partial}{\partial x} (1 0^{x^{2} + x})