RÓWNANIE RÓŻNICZKOWE RZĘDU I O ZMIENNYCH ROZDZIELONYCH
PRZYKŁAD:
$$y^{'} = \frac{y\ln y}{\sin x}$$ $\frac{\text{dx}}{\text{dy}} = \frac{y\ln y}{\sin x}\ \ \ /\ \bullet \frac{\text{dx}}{y}$ $\frac{\text{dy}}{y} = \frac{\ln y}{\sin x}dx\ /\ \bullet \frac{1}{\ln y}$ $\frac{\text{dy}}{y\ln y} = \frac{1}{\sin x}dx\ /\ \bullet \ \int_{}^{}{}$ $$\int_{}^{}\frac{1}{y\ln y}\text{dy} = \int_{}^{}\frac{1}{\sin x}\text{dx}$$ $$\ln\left| \ln y \right| = \ln\left| \text{tg}\frac{x}{2} \right| + C$$ $$\ln\left| \ln y \right| = \ln\left| \text{tg}\frac{x}{2} \right| + \ln C$$ $$e^{\ln\left| \ln y \right|} = e^{\ln\left| \text{tg}\frac{x}{2} \bullet C \right|}$$ $\ln y = \text{tg}\frac{x}{2} \bullet C$ $$y = e^{\text{tg}\frac{x}{2}\ \bullet \ c}$$