(dr(θ))/(r(θ)^2)=(r(θ)dθ)/(θ^2)

Lösung

\frac{d r ( θ )}{r ( θ ) ^{2}} = \frac{r ( θ ) d θ}{θ ^{2}}

θ = - \frac{1}{r} + c_{1}, θ = c_{1}

Schritte zur Lösung

\frac{d r}{r θ ^{2}} = \frac{r d θ}{θ ^{2}}

Lass

θ

die abhängige Variable sein. Teile durch

d r

\frac{\frac{d θ}{d r}}{r θ ^{2}} = \frac{r \frac{d θ}{d r} \cdot \frac{d θ}{d r}}{θ ^{2}}

Ersetze

\frac{d θ}{d r}

mit

θ^{'} (r)

\frac{θ ^{^{'}} ( r )}{r θ ^{2}} = \frac{r θ ^{^{'}} ( r ) θ ^{^{'}} ( r )}{θ ^{2}}

Isoliere

θ^{'} (r) : θ^{'} (r) = \frac{1}{r ^{2}}, θ^{'} (r) = 0

θ^{'} (r) = \frac{1}{r ^{2}} : θ = - \frac{1}{r} + c_{1}

θ^{'} (r) = 0 : θ = c_{1}

θ = - \frac{1}{r} + c_{1}, θ = c_{1}

x y d x + x^{2} d y = 0

\frac{d y}{d x} = \frac{sin ^{2} ( y )}{1 - x ^{2}}

v^{2} x^{2} d x + x^{2} d x - 2 x^{2} v^{2} d x - 2 x^{3} v d v = 0

\frac{d y}{d x} y = 2 3^{x}

x^{^{'}} + 0.09 \cdot x = 0, x (0) = 1