# d^2y/dx^2−2dy/dx+10y=0 where x=0;y=0 and dy/dx=4?x=0;y=0 and dy/dx=4?

$$\displaystyle{d}^{{2}}\frac{{y}}{{\left.{d}{x}\right.}^{{2}}}−{2}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+{10}{y}={0}$$ where $$x=0;y=0$$ and $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={4}?{x}={0};{y}={0}$$ and $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={4}$$?

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Gennenzip

The given differential is $$\displaystyle{d}^{{2}}\frac{{y}}{{\left.{d}{x}\right.}^{{2}}}-{2}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+{10}{y}={0}$$ where $$x=0;y=0$$ and $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={4}$$.
The auxilary equation is given by $$\displaystyle{m}^{{2}}-{2}{m}+{10}={0}$$
$$\rightarrow m=-2+(-\sqrt{4-40}) \rightarrow m=-2\pm2(\sqrt{1-10}) \rightarrow m=-1\pm \sqrt{1-10} \rightarrow m=-1 \pm \sqrt{9} \rightarrow m=-1\pm3i$$
Therefore the general olution is given by $$\displaystyle{y}{\left({x}\right)}={e}^{{-{{x}}}}{\left({c}_{1}{\cos{{\left({3}{x}\right)}}}+{c}_{2}{\sin{{\left({3}{x}\right)}}}\right)}$$
When $$x=0,y=0$$ therefore $$c_{1}=0$$. Therefore $$y=c_2e^{-x}(\sin(3x)) \rightarrow dy/x=c_2e^{-x}(-\sin(3x)+3\cos(3x))$$
Again when $$\displaystyle{x}={0},\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={4}$$, this implies that $$4=3e2$$
Therefore the solution is $$\displaystyle{y}={\left(\frac{{{4}{e}^{{-{{x}}}}}}{{3}}\right)}{\sin{{\left({3}{x}\right)}}}$$