# Question roblem 5. (10 POINTS) Consider the following ODE: $4 t y^{\prime \prime}+2 y^{\prime}+y=0, \quad t>0$ (a) ( 1 point) Show that $$t_{0}=0$$ is a regular singular point of the ODE. (b) (8 points) Use the Frobenius method to construct the general solution of the ODE. Hint: The recurrence relation used in the Frobenius method is: $a_{n} F(r+n)+\sum_{k=0}^{n-1} a_{k}\left[p_{n-k}(k+r)+q_{n-k}\right]=0, \quad n=1,2, \ldots$ (c) (1 point) Find the solution of the ODE that also satisfies the initial conditions $$y(0)=2$$ and $$y^{\prime}(0)=-1$$.

Transcribed Image Text: roblem 5. (10 POINTS) Consider the following ODE: $4 t y^{\prime \prime}+2 y^{\prime}+y=0, \quad t>0$ (a) ( 1 point) Show that $$t_{0}=0$$ is a regular singular point of the ODE. (b) (8 points) Use the Frobenius method to construct the general solution of the ODE. Hint: The recurrence relation used in the Frobenius method is: $a_{n} F(r+n)+\sum_{k=0}^{n-1} a_{k}\left[p_{n-k}(k+r)+q_{n-k}\right]=0, \quad n=1,2, \ldots$ (c) (1 point) Find the solution of the ODE that also satisfies the initial conditions $$y(0)=2$$ and $$y^{\prime}(0)=-1$$.
Transcribed Image Text: roblem 5. (10 POINTS) Consider the following ODE: $4 t y^{\prime \prime}+2 y^{\prime}+y=0, \quad t>0$ (a) ( 1 point) Show that $$t_{0}=0$$ is a regular singular point of the ODE. (b) (8 points) Use the Frobenius method to construct the general solution of the ODE. Hint: The recurrence relation used in the Frobenius method is: $a_{n} F(r+n)+\sum_{k=0}^{n-1} a_{k}\left[p_{n-k}(k+r)+q_{n-k}\right]=0, \quad n=1,2, \ldots$ (c) (1 point) Find the solution of the ODE that also satisfies the initial conditions $$y(0)=2$$ and $$y^{\prime}(0)=-1$$.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/3(a) In the given differential equation $$\mathrm{{p}{\left({t}\right)}=\frac{{1}}{{{2}{t}}},\quad{q}{\left({t}\right)}=\frac{{1}}{{{4}{t}}}}$$both these functions are not analytic at $$\mathrm{{t}={0}}$$But the functions $$\mathrm{{t}{p}{\left({t}\right)}{\quad\text{and}\quad}{t}^{{2}}{q}{\left({t}\right)}}$$ are analytic at $$\mathrm{{t}={0}}$$.Hence $$\mathrm{{t}={0}}$$ is a regular singular point of the ODE.Explanation:Please refer to solution in this step.Step2/3(b) The Frobenius method:$$\mathrm{{y}={\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{t}^{{{n}+{r}}}}$$$$\mathrm{{y}'={\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{\left({n}+{r}\right)}{t}^{{{n}+{r}-{1}}}}$$$$\mathrm{{y}{''}={\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{\left({n}+{r}\right)}{\left({n}+{r}-{1}\right)}{t}^{{{n}+{r}-{2}}}}$$Substituting these in the differential equation:$$\mathrm{{4}{\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{\left({n}+{r}\right)}{\left({n}+{r}-{1}\right)}{t}^{{{n}+{r}-{1}}}+{2}{\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{\left({n}+{r}\right)}{t}^{{{n}+{r}-{1}}}+{\sum_{{{n}={0}}}^{\infty}}{a}_{{n}}{t}^{{{n}+{r}}}={0}}$$\( \mathrm{{\left[{4}{a}_{{0}}{r}{\left({r}-{1}\right)}+{2}{a}_{{0}}{r}\right]}{t}^{{{r}-{1}}}+{\sum_{{{n}={0}}}^{\infty}}{t}^{{{n}+{r}}}{\left[{a}_{{n}}+{4}{a}_{{{n}+{1}}}{\left({n}+{r}+{1}\right)}{\ ... See the full answer