Chapter 5

Verify by direct substitution that the given power series is a particular solution of the indicated differential equation. $$ y=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2^{2 n}(n !)^{2}} x^{2 n}, \quad x y^{\prime \prime}+y^{\prime}+x y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x y^{\prime \prime}-y^{\prime}+2 y=0 $$

In Problems 15 and 16 , without actually solving the given differential equation, find a lower bound for the radius of convergence of power series solutions about the ordinary point \(x=0\). About the ordinary point \(x=1\). $$ \left(x^{2}-25\right) y^{\prime \prime}+2 x y^{\prime}+y=0 $$

$$ y^{\prime \prime}+x y^{\prime}+2 y=0, y(0)=3, y^{\prime}(0)=-2 $$

Without actually solving the given differential equation, find a lower bound for the radius of convergence of power series solutions about the ordinary point \(x=0\). About the ordinary point \(x=1\). $$ \left(x^{2}-25\right) y^{\prime \prime}+2 x y^{\prime}+y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x y^{\prime \prime}+5 y^{\prime}+x y=0 $$

Without actually solving the given differential equation, find a lower bound for the radius of convergence of power series solutions about the ordinary point \(x=0\). About the ordinary point \(x=1\). $$ \left(x^{2}-2 x+10\right) y^{\prime \prime}+x y^{\prime}-4 y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 4 x y^{\prime \prime}+\frac{1}{2} y^{\prime}+y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-3 x y=0 $$

Without actually solving the differential equation $$ (1-2 \sin x) y^{\prime \prime}+x y=0 $$ find a lower bound for the radius of convergence of power series solutions about the ordinary point \(x=0\).

$$ x^{2} y^{\prime \prime}+\left(x^{2}-2\right) y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-3 x y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x^{2} y^{\prime \prime}-x y^{\prime}+\left(x^{2}+1\right) y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+x^{2} y=0 $$

Even though \(x=0\) is an ordinary point of the differential equation, explain why it is not a good idea to try to find a solution of the IVP $$ y^{\prime \prime}+x y^{\prime}+y=0, \quad y(1)=-6, \quad y^{\prime}(1)=3 $$ of the form \(\sum_{n=0}^{\infty} c_{n} x^{n}\). Using power series, find a better way to solve the problem.

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+x^{2} y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 3 x y^{\prime \prime}+(2-x) y^{\prime}-y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-2 x y^{\prime}+y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}-\left(x-\frac{2}{9}\right) y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-x y^{\prime}+2 y=0 $$

$$ 9 x^{2} y^{\prime \prime}+9 x y^{\prime}+\left(x^{6}-36\right) y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-x y^{\prime}+2 y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x y^{\prime \prime}-(3+2 x) y^{\prime}+y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+x^{2} y^{\prime}+x y=0 $$

$$ y^{\prime \prime}+x^{2} y^{\prime}+2 x y=5-2 x+10 x^{3} . $$ Use the assumption \(y=\sum_{n=0}^{\infty} c_{n} x^{n}\) to find the general solution \(y=y_{c}+y_{p}\) that consists of three power series centered at \(x=0\).

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+2 x y^{\prime}+2 y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{4}{9}\right) y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+2 x y^{\prime}+2 y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}+2 x y^{\prime}+2 y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 9 x^{2} y^{\prime \prime}+9 x^{2} y^{\prime}+2 y=0 $$

Express the general solution of the given differential equation on the interval \((0, \infty)\) in terms of Bessel functions. (a) \(4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(64 x^{2}-9\right) y=0\) (b) \(x^{2} y^{\prime \prime}+x y^{\prime}-\left(36 x^{2}+9\right) y=0\)

$$ y^{\prime \prime}+y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ (x-1) y^{\prime \prime}+y^{\prime}=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x^{2} y^{\prime \prime}+3 x y^{\prime}+(2 x-1) y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ (x+2) y^{\prime \prime}+x y^{\prime}-y=0 $$

$$ x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x^{2}+2\right) y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ (x+2) y^{\prime \prime}+x y^{\prime}-y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x y^{\prime \prime}+2 y^{\prime}-x y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-(x+1) y^{\prime}-y=0 $$

Use binomial series to formally show that $$ \left(1-2 x t+t^{2}\right)^{-1 / 2}=\sum_{n=0}^{\infty} P_{n}(x) $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ y^{\prime \prime}-(x+1) y^{\prime}-y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ \left(x^{2}+1\right) y^{\prime \prime}-6 y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ \left(x^{2}+1\right) y^{\prime \prime}-6 y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x y^{\prime \prime}-x y^{\prime}+y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ \left(x^{2}+2\right) y^{\prime \prime}+3 x y^{\prime}-y=0 $$

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ \left(x^{2}+2\right) y^{\prime \prime}+3 x y^{\prime}-y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ y^{\prime \prime}+\frac{3}{x} y^{\prime}-2 y=0 $$

In Problems \(17-28\), find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ \left(x^{2}-1\right) y^{\prime \prime}+x y^{\prime}-y=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x y^{\prime \prime}+(1-x) y^{\prime}-y=0 $$