Chapter 5

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x^{3} y^{\prime \prime}+4 x^{2} y^{\prime}+3 y=0 $$

In Problems \(1-4\), find the radius of convergence and interval of convergence for the given power series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n} x^{n} $$

The general solution of \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) is \(y=c_{1} J_{1}(x)+c_{2} J_{-1}(x)\).

Find the radius of convergence and interval of convergence for the given power series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n} x^{n} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x(x+3)^{2} y^{\prime \prime}-y=0 $$

In Problems, use (1) to find the general solution of the given differential equation on \((0, \infty)\). $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0 $$

In Problems \(1-4\), find the radius of convergence and interval of convergence for the given power series. $$ \sum_{n=0}^{\infty} \frac{(100)^{n}}{n !}(x+7)^{n} $$

Since \(x=0\) is an irregular singular point of \(x^{3} y^{\prime \prime}-x y^{\prime}+\) \(y=0\), the DE possesses no solution that is analytic at \(x=0\).

Find the radius of convergence and interval of convergence for the given power series. $$ \sum_{n=0}^{\infty} \frac{(100)^{n}}{n !}(x+7)^{n} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ \left(x^{2}-9\right)^{2} y^{\prime \prime}+(x+3) y^{\prime}+2 y=0 $$

In Problems, use (1) to find the general solution of the given differential equation on \((0, \infty)\). $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}-25\right) y=0 $$

In Problems \(1-4\), find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=1}^{\infty} \frac{(-1)^{k}}{\underline{10^{k}}}(x-5)^{k} $$

Both power series solutions of \(y^{\prime \prime}+\ln (x+1) y^{\prime}+y=0\) centered at the ordinary point \(x=0\) are guaranteed to converge for all \(x\) in which one of the following intervals? (a) \((-\infty, \infty)\) (b) \((-1, \infty)\) (c) \(\left[-\frac{1}{2}, \frac{1}{2}\right]\) (d) \([-1,1]\)

Find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=1}^{\infty} \frac{(-1)^{k}}{10^{k}}(x-5)^{k} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ y^{\prime \prime}-\frac{1}{x} y^{\prime}+\frac{1}{(x-1)^{3}} y=0 $$

In Problems, use (1) to find the general solution of the given differential equation on \((0, \infty)\). $$ 16 x^{2} y^{\prime \prime}+16 x y^{\prime}+\left(16 x^{2}-1\right) y=0 $$

In Problems \(1-4\), find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=0}^{\infty} k !(x-1)^{k} $$

\(x=0\) is an ordinary point of a certain linear differential equation. After the assumed solution \(y=\sum_{n=0}^{\infty} c_{n} x^{n}\) is substituted into the \(\mathrm{DE}\), the following algebraic system is obtained by equating the coefficients of \(x^{0}, x^{1}, x^{2}\), and \(x^{3}\) to zero: $$ \begin{aligned} 2 c_{2}+2 c_{1}+c_{0} &=0 \\ 6 c_{3}+4 c_{2}+c_{1} &=0 \\ 12 c_{4}+6 c_{3}+c_{2}-\frac{1}{3} c_{1} &=0 \\ 20 c_{5}+8 c_{4}+c_{3}-\frac{2}{3} c_{2} &=0 \end{aligned} $$ Bearing in mind that \(c_{0}\) and \(c_{1}\) are arbitrary, write down the first five terms of two power series solutions of the differential equation.

Find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=0}^{\infty} k !(x-1)^{k} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ \left(x^{3}+4 x\right) y^{\prime \prime}-2 x y^{\prime}+6 y=0 $$

Suppose the powers series \(\sum_{n=0} c_{k}(x-4)^{k}\) is known to converge at \(-2\) and diverge at 13 . Discuss whether the series converges at \(-7,0,7,10\), and 11 . Possible answers are does, does not, or might.

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x^{2}(x-5)^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x^{2}-25\right) y=0 $$

In Problems 5 and 6 , the given function is analytic at \(x=0\). Find the first four terms of a power series in \(x\). Perform the multiplication by hand or use a CAS, as instructed. $$ e^{-x} \cos x $$

Jse the Maclaurin series for \(\sin x\) and \(\cos x\) along with long livision to find the first three nonzero terms of a power series n \(x\) for the function \(f(x)=\frac{\sin x}{\cos x}\).

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ \left(x^{2}+x-6\right) y^{\prime \prime}+(x+3) y^{\prime}+(x-2) y=0 $$

In Problems 7 and 8 , the given function is analytic at \(x=0\). Find the first four terms of a power series in \(x\). Perform the long division by hand or use a CAS, as instructed. Give the open interval of convergence. $$ \frac{1}{\cos x} $$

The given function is analytic at \(x=0\). Find the first four terms of a power series in \(x\). Perform the long division by hand or use a CAS, as instructed. Give the open interval of convergence. $$ \frac{1}{\cos x} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x\left(x^{2}+1\right)^{2} y^{\prime \prime}+y=0 $$

In Problems 7 and 8 , the given function is analytic at \(x=0\). Find the first four terms of a power series in \(x\). Perform the long division by hand or use a CAS, as instructed. Give the open interval of convergence. $$ \frac{1-x}{2+x} $$

The given function is analytic at \(x=0\). Find the first four terms of a power series in \(x\). Perform the long division by hand or use a CAS, as instructed. Give the open interval of convergence. $$ \frac{1-x}{2+x} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x^{3}\left(x^{2}-25\right)(x-2)^{2} y^{\prime \prime}+3 x(x-2) y^{\prime}+7(x+5) y=0 $$

Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ 2 x y^{\prime \prime}+y^{\prime}+y=0 $$

Rewrite the given power series so that its general term involves \(x^{k}\). $$ \sum_{n=1}^{\infty} n c_{n} x^{n+2} $$

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ \left(x^{3}-2 x^{2}+3 x\right)^{2} y^{\prime \prime}+x(x-3)^{2} y^{\prime}-(x+1) y=0 $$

In Problems 9 and 10 , rewrite the given power series so that its general term involves \(x^{k}\) $$ \sum_{n=3}^{\infty}(2 n-1) c_{n} x^{n-3} $$

Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ y^{\prime \prime}-x y^{\prime}-y=0 $$

Rewrite the given power series so that its general term involves \(x^{k}\). $$ \sum_{n=3}^{\infty}(2 n-1) c_{n} x^{n-3} $$

Put the given differential equation into the form (3) for each regular singular point of the equation. Identify the functions \(p(x)\) and \(q(x)\). $$ \left(x^{2}-1\right) y^{\prime \prime}+5(x+1) y^{\prime}+\left(x^{2}-x\right) y=0 $$

In Problems, use the indicated change of variable to find the general solution of the given differential equation on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}+2 x y^{\prime}+\alpha^{2} x^{2} y=0 ; \quad y=x^{-1 / 2} u(x) $$

Rewrite the given expression as a single power series whose general term involves \(x^{k}\). $$ \sum_{\substack{n=1 \\ \infty}}^{\infty} 2 n c_{n} x^{n-1}+\sum_{n=0}^{\infty} 6 c_{n} x^{n+1} $$

Put the given differential equation into the form (3) for each regular singular point of the equation. Identify the functions \(p(x)\) and \(q(x)\). $$ x y^{\prime \prime}+(x+3) y^{\prime}+7 x^{2} y=0 $$

In Problems, use the indicated change of variable to find the general solution of the given differential equation on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-\nu^{2}+\frac{1}{4}\right) y=0 ; \quad y=\sqrt{x} u(x) $$

Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ y^{\prime \prime}-x^{2} y^{\prime}+x y=0 $$

$$ x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-\nu^{2}+\frac{1}{4}\right) y=0 ; \quad y=\sqrt{x} u(x) $$

Rewrite the given expression as a single power series whose general term involves \(x^{k}\). $$ \sum_{n=2}^{\infty} n(n-1) c_{n} x^{n}+2 \sum_{n=2}^{\infty} n(n-1) c_{n} x^{n-2}+3 \sum_{n=1}^{\infty} n c_{n} x^{n} $$

x=0$ is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving, discuss the number of series solutions you would expect to find using the method of Frobenius. $$ x^{2} y^{\prime \prime}+\left(\frac{5}{3} x+x^{2}\right) y^{\prime}-\frac{1}{3} y=0 $$

Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ x y^{\prime \prime}-(x+2) y^{\prime}+2 y=0 $$

$$ x^{2} y^{\prime \prime}+\left(\frac{5}{3} x+x^{2}\right) y^{\prime}-\frac{1}{3} y=0 $$

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

In Problems 13 and 14 , verify by direct substitution that the given power series is a particular solution of the indicated differential equation. $$ y=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2^{2 n}(n !)^{2}} x^{2 n}, \quad x y^{\prime \prime}+y^{\prime}+x y=0 $$