91Ó°ÊÓ

In Exercises 14-25 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ 2 x^{2} y^{\prime \prime}+5 x y^{\prime}+(1+x) y=0 $$

Find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{n+r_{1}}\), where \(0 \leq n \leq N(N\) at least 7\()\) and \(r_{1}\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(N>7\). \(x^{2}(1+2 x) y^{\prime \prime}+x(5+9 x) y^{\prime}+(4+3 x) y=0\)

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ x^{2} y^{\prime \prime}-x\left(7-x^{2}\right) y^{\prime}+12 y=0 $$

Find \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the power series \(y=\sum_{n=0}^{\infty} a_{n}\left(x-x_{0}\right)^{n}\) for the solution of the initial value problem. Take \(x_{0}\) to be the point where the initial conditions are imposed. $$ \left(4 x^{2}-24 x+37\right) y^{\prime \prime}+y=0, \quad y(3)=4, \quad y^{\prime}(3)=-6 $$

In Exercises 14-25 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients in each solution. $$ x^{2}(3+4 x) y^{\prime \prime}+x(5+18 x) y^{\prime}-(1-12 x) y=0 $$

Find \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the power series \(y=\sum_{n=0}^{\infty} a_{n}\left(x-x_{0}\right)^{n}\) for the solution of the initial value problem. Take \(x_{0}\) to be the point where the initial conditions are imposed. $$ \left(x^{2}-8 x+14\right) y^{\prime \prime}-8(x-4) y^{\prime}+20 y=0, \quad y(4)=3, \quad y^{\prime}(4)=-4 $$

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ x y^{\prime \prime}-5 y^{\prime}+x y=0 $$

Find a fundamental set of Frobenius solutions. Compute the terms involving \(x^{n+r_{1}}\), where \(0 \leq n \leq N(N\) at least 7\()\) and \(r_{1}\) is the root of the indicial equation. Optionally, write a computer program to implement the applicable recurrence formulas and take \(N>7\). \(x^{2}(1+x) y^{\prime \prime}+x(1+2 x) y^{\prime}+x y=0\)

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ x^{2} y^{\prime \prime}+x\left(1+2 x^{2}\right) y^{\prime}-\left(1-10 x^{2}\right) y=0 $$

Find \(a_{0}, \ldots, a_{N}\) for \(N\) at least 7 in the power series \(y=\sum_{n=0}^{\infty} a_{n}\left(x-x_{0}\right)^{n}\) for the solution of the initial value problem. Take \(x_{0}\) to be the point where the initial conditions are imposed. $$ \left(2 x^{2}+4 x+5\right) y^{\prime \prime}-20(x+1) y^{\prime}+60 y=0, \quad y(-1)=3, \quad y^{\prime}(-1)=-3 $$