Series Solutions of Differential Equations

In problems 1-6, determine the convergence set of the given power series.

 $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{n}}^{\mathbf{2}}}{{\mathbf{2}}^{\mathbf{n}}}{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{n}}$

In problems 1-6, determine the convergence set of the given power series.

  $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{3}}{{\mathbf{n}}^{\mathbf{3}}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}^{\mathbf{n}}$

Duffing's Equation. In the study of a nonlinear spring with periodic forcing, the following equation arises:

${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathit{k}\mathit{y}\mathbf{+}\mathit{r}{\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}$

Let $\mathit{k}\mathbf{=}\mathit{r}\mathbf{=}\mathit{A}\mathbf{=}\mathbf{1}$and.$\mathit{\omega }\mathbf{=}\mathbf{10}$Find the first three nonzero terms in the Taylor polynomial approximations to the solution with initial values .$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$

For Duffing's equation given in Problem 13, the behaviour of the solutions changes as r changes sign. When $\mathit{r}\mathbf{>}\mathbf{0}$ , the restoring force $\mathit{k}\mathit{y}\mathbf{+}\mathit{r}{\mathit{y}}^{\mathbf{3}}$ becomes stronger than for the linear spring$\mathbf{(}\mathit{r}\mathbf{=}\mathbf{0}\mathbf{)}$. Such a spring is called hard. When $\mathit{r}\mathbf{<}\mathbf{0}$, the restoring force becomes weaker than the linear spring and the spring is called soft. Pendulums act like soft springs.

(a) Redo Problem 13 with $\mathit{r}\mathbf{=}\mathbf{-}\mathbf{1}$. Notice that for the initial conditions ,$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$ the soft and hard springs appear to respond in the same way for small.

(b) Keeping $\mathit{k}\mathbf{=}\mathit{A}\mathbf{=}\mathbf{1}$ and ,$\mathit{\omega }\mathbf{=}\mathbf{10}$ change the initial conditions to $\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$and ${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$. Now redo Problem 13 with $\mathit{r}\mathbf{=}\mathbf{\pm }\mathbf{1}$.

(c) Based on the results of part (b), is there a difference between the behavior of soft and hard springs for small? Describe.

The solution to the initial value problem

$\begin{array}{l}x{y}^{\text{'}\text{'}}\left(x\right)+2y^{\text{'}}\left(x\right)+xy\left(x\right)=0\\ y\left(0\right)=1,y^{\text{'}}\left(0\right)=0\end{array}$

has derivatives of all orders at $\mathit{x}\mathbf{=}\mathbf{0}$ (although this is far from obvious). Use L'Hôpital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

In the study of the vacuum tube, the following equation is encountered:

${\mathit{y}}^{\mathbf{"}}\mathbf{+}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}\mathbf{(}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{.}$

Find the Taylor polynomial of degree 4 approximating the solution with the initial values$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$ , ${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

In problems 1-6, determine the convergence set of the given power series.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{2}}^{\mathbf{-}\mathbf{n}}}{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}$

In problems 1-6, determine the convergence set of the given power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{3}}^{\mathbf{n}}}{\mathbf{n}\mathbf{!}}{\mathit{x}}^{\mathbf{n}}$

In problems 1-6, determine the convergence set of the given power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{4}}{{\mathbf{n}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{n}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{)}}^{\mathbf{n}}$

In problems 1-6, determine the convergence set of the given power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{!}}{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{n}}$

Question:7. Sometimes the ratio test (Theorem 2) can be applied to a power series containing an infinite number of zero coefficients, provided the zero pattern is regular. Use Theorem 2 to show, for example, that the series

has a radius of convergence  , if

and that

has a radius of convergence , if

Question : find the power series expansion _for given the expansions for f(x) and g(x).

Question:In Problem find the first three nonzero terms in the power series expansion for the product f(x)g(x).

Question: To find the first few terms in the power series for the quotient q(x) in Problem 15, treat the power series in the numerator and denominator as "long polynomials" and carry out long division. That is, perform

16. $1+x+\frac{1}{2}+...$ 

Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x) .

17. $f\left(x\right)={\left(1+x\right)}^{-1}=\sum _{n=0}^m(-1^n{x}^n$

Question: In Problems 29–34, determine the Taylor series about the point x₀ for the given functions and values of x₀.

29. f(x)= cosx , x_{0 = $\pi$}

Question: In Problems 29–34, determine the Taylor series about the point X₀  for the given functions and values of X₀.

30.  $f\left(x\right)=x^{-4} x_0=1$ 

Question: In Problems 29–34, determine the Taylor series about the point X₀  for the given functions and values of X₀.

32. f(x)=ln(1+x), x₀ =0  

Question: In Problems 29–34, determine the Taylor series about the point X₀ for the given functions and values of X₀ .

33. f (X)= x³+3x-4, x₀ = 1

Question: In Problems 29–34, determine the Taylor series about the point x₀  for the given functions and values of x₀ .

34. f(x)=$\sqrt{x,} x_0=1$

Question: Let   

Show that fⁿ(0)=0 for n=0,1,2....  and hence that the Maclaurin series for f(x)  is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x₀ =0 ), whose Taylor series converges, but the Taylor series (about x₀ =0) does not converge to the original function! Consequently, this function is not analytic at x=0.

Question: Compute the Taylor series for f(x)= in(1+x²) about x₀ = 0. [Hint: Multiply the series for (1+x²)^-1 by 2x and integrate.]

Question: In Problems 1–10, determine all the singular points of the given differential equation.

1. (x+1)y"-x²y'+3y = 0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

2. x²y"-3y-xy = 0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

3. $\left({\theta }^2-2\right)y"+2y\text{'}+\left(\sin \theta \right)y=0$

Question: In Problems 1–10, determine all the singular points of the given differential equation.

4. (x²+x)y"+3y'-6xy = 0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

5. (t² - t -2)x" + (t +1)x' - (t - 2)x = 0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

6 (x² - 1)y" + (1 - x)y' + (x² - 2x + 1)y = 0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

7. (sinx)y"+(cosx)y =0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

8. e^xy"-(x²-1)y'+2xy=0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

9. (sin)y"-(in)y=0

Use Table 6.4.1 to find the first three positive eigen values and corresponding eigen functions of the boundary-value problem\(xy'' + y' + \lambda xy = 0,y(x),y'(x)\)bounded as \(x \to {0^ + },y(2) = 0\). (Hint: By identifying \(\lambda   = {\alpha ^2}\), the DE is the parametric Bessel equation of order zero.)

(a) Use (20) to show that the general solution of the differential equation \(xy'' + \lambda y = 0\) on the interval \((0,\infty )\) is\(y = {c_1}\sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right) + {c_2}\sqrt x {Y_1}\left( {2\sqrt {\lambda x} } \right)\).

(b) Verify by direct substitution that \(y = \sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right)\)is a particular solution of the DE in the case \(\lambda  = 1\).

Use the result in parts (a) and (b) of Problem 36 to express the general solution on \((0,\infty )\) of each of the two forms of Airy’s equation in terms of Bessel functions.

In Problems 29 and 30 use (22) or (23) to obtain the given result.

\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)

Show that \(y = {x^{1/2}}w\left( {\frac{2}{3}\alpha {x^{3/2}}} \right)\)is a solution of the given form of Airy’s differential equation whenever w is a solution ofthe indicated Bessel’s equation. (Hint: After differentiating, substituting, and simplifying, then let \(t = \frac{2}{3}\alpha {x^{3/2}}\))

(a) \(y'' + {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' + \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)

(b)\(y'' - {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' - \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)

Use the change of variables \(s = \frac{2}{\alpha }\sqrt {\frac{k}{m}} {e^{ - \alpha t/2}}\)to show that the differential equation of the aging spring \(mx'' + k{e^{ - \alpha t}}x = 0\),\(\alpha   > 0\)becomes\({s^2}\frac{{{d^2}x}}{{d{s^2}}} + s\frac{{dx}}{{ds}} + {s^2}x = 0\).

(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).

(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.

Find at least the first four non-zero terms in a power series expansion about x₀ for a general solution to the given differential equation with the value for x₀.

$\left(x^2-2x\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }{\mathbf{x}}_{\mathbf{0}}\mathbf{=}\mathbf{1}$

Find at least the first four nonzero terms in a power series expansion about x₀ for a general solution to the given differential equation with the given value for x₀.

${\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{xy}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }{\mathbf{x}}_{\mathbf{0}}\mathbf{=}\mathbf{2}$

Find at least the first four nonzero terms in a power series expansion about ${\mathbf{x}}_{\mathbf{0}}$ for a general solution to the given differential equation with the given value for ${\mathbf{x}}_{\mathbf{0}}$,

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{(}\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }{\mathbf{x}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathbf{1}$

Find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem, $\mathbf{x}\mathbf{\text{'}}\mathbf{+}\left(\sin t\right)\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{-}{\mathbf{e}}^{\mathbf{x}}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$

In Problems 13-19, find at least the first four non-zero terms in a power series expansion of the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}{\mathbf{e}}^{\mathbf{t}}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{\left(}\mathbf{sinx}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{y}\mathbf{\left(}\mathbf{\pi }\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{\pi }\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0.

x²y"(x)+6xy'(x)+6y(x)=0

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0. 

2x²y"(x)+13xy'(x)+15y(x)=0

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0.

x²y"+xy'(x)+17y=0

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0. 

x²y"+2xy'-3y=0