(ux)^2udx+xdu+x^2+(x)(ux)+(ux)^2=0

Lösung

(ux)^{2} u d x + x d u + x^{2} + (x) (ux) + (ux)^{2} = 0

x (u) = \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}, x (u) = - \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}

Schritte zur Lösung

(ux (u))^{2} u d x (u) + x (u) d u + x (u)^{2} + (x (u)) (ux (u)) + (ux (u))^{2} = 0

Solve separable ODE:

x (u) = \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}, x (u) = - \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}

x (u) = \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}, x (u) = - \frac{- 2 ln ( u + 1 ) + ln ( u ^{2} + 1 ) - 2 arctan ( u ) + c _{1}}{2}

\frac{d r}{d t} = - r

y = x \frac{d y}{d x}

y^{^{'}} - 2 e^{x} y = - 2 e^{x}

y^{^{'}} = \frac{x ( x ^{2} + 1 )}{( 4 y ^{2} )}

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