RÓWNANIE RÓŻNICZKOWE LINIOWE – METODA PRZEWIDYWAŃ
PRZYKŁAD
y^′ + 3y = 0
$$\frac{\text{dy}}{\text{dx}} = - 3y\ \ / \bullet \frac{\text{dx}}{y}$$
$$\frac{\text{dy}}{y} = - 3dx\ \ / \bullet \int_{}^{}{}$$
$$\int_{}^{}\frac{1}{y}\text{dy} = - 3\int_{}^{}\text{dx}$$
ln|y| = −3x + C
y = e^−3x + C
CORLJ
y^′ + 3y = xe^3x
y^′ = Ae^3x + (Ax+B)e^3x  • 3
$$Ae^{3x} + \left( 3Ax + 3B \right)e^{3x} + \left( 3Ax + 3B \right)e^{3x} = xe^{3x}\ \ / \bullet \frac{1}{e^{3x}}$$
A + 3Ax + 3B + 3Ax + 3B = x
$$\begin{matrix} x^{1} \\ x^{0} \\ \end{matrix}\left\{ \begin{matrix} 1 = 3A + 3A \\ 0 = A + 3B + 3B \\ \end{matrix} \right.\ $$ $$A = \frac{1}{6}$$ $$B = - \frac{1}{36}$$
$$y = \left( \frac{1}{6}x - \frac{1}{36} \right)e^{3x}$$
$$y = C_{1}e^{- 3x} + \frac{1}{6}\left( x - \frac{1}{6} \right)e^{3x}$$