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Question Number 95447 by Tony Lin last updated on 25/May/20

$$canIwritethesolutionof$$$${ay}^{″}+{by}^{\prime }+cy=0$$$$y=\{\begin{array}{l}{c}_1{e}^{\frac{-b+\sqrt{b^2-4ac}}{2}x}+c_2{e}^{\frac{-b-\sqrt{b^2-4ac}}{2}x},when{b}^2-4ac\ne 0\\ c_1{e}^{\frac{-b}{2}x}+c_2{xe}^{\frac{-b}{2}x},when{b}^2-4ac=0\end{array}$$$$inonesentence$$$$notintheformofpiecewide-definefunction$$

Answered by mathmax by abdo last updated on 25/May/20

$$\mathrm{the}\mathrm{caracteristic}\mathrm{equation}\mathrm{is}{\mathrm{ar}}^2+\mathrm{br}+\mathrm{c}=0$$$$\mathrm{\Delta }={\mathrm{b}}^2-4\mathrm{ac}\mathrm{case}1\to {\mathrm{b}}^2-4\mathrm{ac}>0\Rightarrow {\mathrm{r}}_1=\frac{-\mathrm{b}+\sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$$$$\mathrm{and}{\mathrm{r}}_2=\frac{-\mathrm{b}-\sqrt{{\mathrm{b}}^2-4\mathrm{ax}}}{2\mathrm{a}}\Rightarrow \mathrm{the}\mathrm{solution}\mathrm{is}\mathrm{y}\left(\mathrm{x}\right)=\alpha {\mathrm{e}}^{{\mathrm{r}}_1\mathrm{x}}+\beta {\mathrm{e}}^{{\mathrm{r}}_2\mathrm{x}}$$$$\mathrm{case}2\to {\mathrm{b}}^2-4\mathrm{ac}<0\Rightarrow {\mathrm{r}}_1=\frac{-\mathrm{b}+\mathrm{i}\sqrt{4\mathrm{ac}-{\mathrm{b}}^2}}{2\mathrm{a}}\mathrm{and}{\mathrm{r}}_2=\frac{-\mathrm{b}-\mathrm{i}\sqrt{4\mathrm{ac}-{\mathrm{b}}^2}}{2\mathrm{a}}$$$$\Rightarrow \mathrm{y}\left(\mathrm{x}\right)={\mathrm{c}}_1{\mathrm{e}}^{{\mathrm{r}}_1\mathrm{x}}+{\mathrm{c}}_2{\mathrm{e}}^{{\mathrm{r}}_2\mathrm{x}}$$$$\mathrm{case}3\to {\mathrm{b}}^2-4\mathrm{ac}=0\Rightarrow \mathrm{one}\mathrm{roots}\mathrm{r}=\frac{-\mathrm{b}}{2\mathrm{a}}\Rightarrow$$$$\mathrm{y}\left(\mathrm{x}\right)=(\mathrm{ax}+\mathrm{b}){\mathrm{e}}^{\mathrm{rx}}$$

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