Chapter 8: Series Solutions of Differential Equations

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)$y^{″}+{\alpha }^{2}xy=0,x>0;t^2{w}^{″}+t{w}^{\prime }+\left(t^2-\frac{1}{9}w\right)=0,t>0$

(b)$y^{″}-{\alpha }^{2}xy=0,x>0;t^2{w}^{″}+t{w}^{\prime }-\left(t^2-\frac{1}{9}w\right)=0,t>0$

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 $m{x}^{″}+k{e}^{-\alpha t}x=0$,$\alpha >0$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 a minimum value for the radius of convergence of a power series solution about x₀.

(x²-5x+6) y"-3xy'-y=0; x₀=0

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

8. e^xy"-(x²-1)y'+2xy=0

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

x³y"'+3x²y"+5xy'-5y=0

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

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

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

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.

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