91Ó°ÊÓ

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises \(5-26\). $$ x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+\left(x^{2}-\frac{1}{9}\right) y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ \left(x^{3}-1\right) \frac{d^{2} y}{d x^{2}}+x^{2} \frac{d y}{d x}+x y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises \(1-10 .\) $$ (x+3) \frac{d^{2} y}{d x^{2}}+(x+2) \frac{d y}{d x}+y=0 $$

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises \(5-26\). $$ 2 x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+2 y=0 $$

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises \(5-26\). $$ 3 x \frac{d^{2} y}{d x^{2}}-(x-2) \frac{d y}{d x}-2 y=0 $$

Find the power series solution of each of the initial-value problems in Exercises \(11-14 .\) $$ \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 $$

Find the power series solution of each of the initial-value problems in Exercises \(11-14 .\) $$ \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-2 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1 $$

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises \(5-26\). $$ x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+x y=0 $$

Find the power series solution of each of the initial-value problems in Exercises \(11-14 .\) $$ \left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+2 x y=0, \quad y(0)=2, \quad y^{\prime}(0)=3 $$

Find the power series solution of each of the initial-value problems in Exercises \(11-14 .\) $$ \left(2 x^{2}-3\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+y=0, \quad y(0)=-1, \quad y^{\prime}(0)=5 $$