# Question Problem 4. (10 POINTS) Consider the following IVP: $\left\{\begin{array}{l} y^{\prime \prime}-2 t y^{\prime}+8 y=0 \\ y(0)=4, y^{\prime}(0)=0 \end{array}\right.$ (a) ( 1 point) Is $$t_{0}=0$$ an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. (b) (8 points) Use the power series approach to find the solution of the IVP. (c) (1 point) What is the radius of convergence of the resulting power series?

Transcribed Image Text: Problem 4. (10 POINTS) Consider the following IVP: $\left\{\begin{array}{l} y^{\prime \prime}-2 t y^{\prime}+8 y=0 \\ y(0)=4, y^{\prime}(0)=0 \end{array}\right.$ (a) ( 1 point) Is $$t_{0}=0$$ an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. (b) (8 points) Use the power series approach to find the solution of the IVP. (c) (1 point) What is the radius of convergence of the resulting power series?
Transcribed Image Text: Problem 4. (10 POINTS) Consider the following IVP: $\left\{\begin{array}{l} y^{\prime \prime}-2 t y^{\prime}+8 y=0 \\ y(0)=4, y^{\prime}(0)=0 \end{array}\right.$ (a) ( 1 point) Is $$t_{0}=0$$ an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. (b) (8 points) Use the power series approach to find the solution of the IVP. (c) (1 point) What is the radius of convergence of the resulting power series?
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/2y''-2ty'+8y=0put y=$$\mathrm{\Sigma}$$an $$\mathrm{{t}^{{n}}}$$and y'=n$$\mathrm{\Sigma}$$an$$\mathrm{{t}^{{{n}-{1}}}}$$and y''=n(n-1)$$\mathrm{\Sigma{a}{n}{t}^{{{n}-{2}}}}$$ put these values in IVPWe get,n(n-1)$$\mathrm{\Sigma{a}{n}{t}^{{{n}-{2}}}}$$-2t$$\mathrm{\Sigma{a}{n}{t}^{{{n}-{1}}}}$$ +8$$\mathrm{\Sigma{a}{n}{t}^{{n}}}$$=0n(n-1)$$\mathrm{\Sigma{a}{n}{t}^{{{n}-{2}}}}$$-2$$\mathrm{\Sigma{a}{n}{t}^{{{n}}}}$$ +8$$\mathrm{\Sigma{a}{n}{t}^{{n}}}$$=0n(n-1)$$\mathrm{\Sigma{a}{n}{t}^{{{n}-{2}}}=-{6}\Sigma{a}{n}{t}^{{n}}}$$n(n-1)\( \mathrm ... See the full answer