Series Solutions of Differential Equations

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

d²y/dx²=5/x dy/dx-13/x² y

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

d²y/dx²=1/x dy/dx-4/x² y

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

x³y"'+4x²y"+10xy'-10y=0

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

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.

$$\begin{gathered} \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{\left(}\mathbf{cosx}\mathbf{\right)}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0} \\ \mathbf{y}\mathbf{(}\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{(}\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{1} \end{gathered}$$

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.

$$\begin{gathered} \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{\left(}\mathbf{cosx}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{0} \\ \mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1} \end{gathered}$$

To derive the general solutions given by equations (17)- (20)  for the non-homogeneous equation (16), complete the following steps.

(a) Substitute $\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}{\mathbf{a}}_{\mathbf{n}}{\mathbf{x}}^{\mathbf{n}}$ and the Maclaurin series into equation (16) to obtain 

$(2a_2-a_0)\mathbf{+}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\left[(k+2)(k+1)a_{k+2}-(k+1)a_k\right]{\mathbf{x}}^{\mathbf{k}}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}}{\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{!}}{\mathbf{x}}^{\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}}$

(b) Equate the coefficients of like powers on both sides of the equation in part (a) and thereby deduce the equations

${\mathbf{a}}_{\mathbf{2}}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{,}{\mathbf{a}}_{\mathbf{3}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}}\mathbf{+}\frac{{\mathbf{a}}_{\mathbf{1}}}{\mathbf{3}}\mathbf{,}{\mathbf{a}}_{\mathbf{4}}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{8}}\mathbf{,}{\mathbf{a}}_{\mathbf{5}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{40}}\mathbf{+}\frac{{\mathbf{a}}_{\mathbf{1}}}{\mathbf{15}}\mathbf{,}{\mathbf{a}}_{\mathbf{6}}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{48}}\mathbf{,}{\mathbf{a}}_{\mathbf{7}}\mathbf{=}\frac{\mathbf{19}}{\mathbf{5040}}\mathbf{+}\frac{{\mathbf{a}}_{\mathbf{1}}}{\mathbf{105}}$ 

(c) Show that the relations in part (b) yield the general solution to (16) given in equations (17)-(20).

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about x=0 of a general solution to the given differential equation. 

y'-xy=sinx

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four non-zero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

w'+xw=e^x

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

z"+xz'+z=x²+2x+1

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion abouts x=0 of a general solution to the given differential equation.

y"2xy'+3y=x²

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

(1+x²)y"-xy'+y=e^-x

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation. 

y"-xy'+2y=cosx

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

(1-x²) y"-y'+y=tan x

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation. 

y"-(sin x)y=cos x

The equation

(1-x²)y"-2xy'+n(n+1)y=0

where n is an unspecified parameter is called Legendre’s equation. This equation appears in applications of differential equations to engineering systems in spherical coordinates.

(a) Find a power series expansion about x=0 for a solution to Legendre’s equation. 

(b) Show that for a non negative integer there exists an n^th degree polynomial that is a solution to Legendre’s equation. These polynomials upto a constant multiples are called Legendre polynomials.

(c) Determine the first three Legendre polynomials (upto a constant multiple).

Aging spring. As a spring ages, its “spring constant” decreases on value. One such model for a mass-spring system with an aging spring is mx"(t)+bx'(t)+ke^{- ηt}x(t)=0 .

Where m is the mass, b the damping constant, k and η  positive constants and x(t) displacement of the spring from equilibrium position. Let m=1 kg, b=2 Nsec/m, k=1 N/m, η =1 sec^-1. The system is set in motion by displacing the mass 1m from it equilibrium position and releasing it (x(0)=1, x'(0)=0). Find at least the first four nonzero terms in a power series expansion of about t=0 of displacement.

Aging spring without damping. In a mass-spring system of aging spring discussed in Problem 30, assume that there is no damping (i.e., b=0), m=1 and k=1. To see the effect of aging consider as positive parameter.

(a) Redo Problem 30 with b=0 and η arbitrary but fixed. 

(b) Set η =0 in the expansion obtained in part (a). Does this expansion agree with the expansion for the solution to the problem with η=0. [Hint: When η =0 the solution is x(t)=cos t].

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

x³y"'+3x²y"+5xy'-5y=0

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

x³y"'+9x²y"+19xy'+8y=0

 In Problems 11 and 12, use a substitution of the form to find a general solution to the given equation for x>c.

2(x-3)² y"+ 5(x-3)y'-2y=0

In Problems 11 and 12, use a substitution of the form to find a general solution to the given equation for x>c.

4(x+2)²y"+5y=0

In Problems 13 and 14, use variation of parameters to find a general solution to the given equation for x>0.

x²y"(x)-2xy'(x)+2y(x)=x^-1/2

In Problems 13 and 14, use variation of parameters to find a general solution to the given equation for x>0.

x²y"(x)+2xy'(x)-2y(x)=6x^-2+3x

In Problems 15-17, solve the given initial value problem t²x"-12x=0. x(1)=3 and x'(1)=5.

In Problems 15-17,solve the given initial value problem x²y"+5xy'+4y=0.

y(1) =3 and y'(1) = 7

In Problems 15-17,solve the given initial value problem x³y"'+6x²y"+29xy'-29y=0  y(1)=1 and y'(1)= -3 and y"(1)=19.

Suppose r₀ is a repeated root of the auxiliary equation ar²+br+c=0. Then, as we well know, is a solution to the equation ay"+by'+cy=0 where a, b, and c are constants. Use a derivation similar to the one given in this section for the case when the indicial equation has a repeated root to show that a second linearly independent solution is y₂ (t)=te^rt .

In Problems \(5 - 14\) solve the given linear system.

\({\bf{X'}} = \left( {\begin{array}{*{20}{c}}{{\rm{   0      2      1}}}\\{1{\rm{      }}1{\rm{    }} - 2}\\{2{\rm{       }}2{\rm{   }} - 1}\end{array}} \right){\bf{X}}\)