2.5.14 problem 14

Internal problem ID [8401]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 14
Date solved : Sunday, November 10, 2024 at 03:44:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\frac {1}{x} \end{align*}

Using series expansion around \(x=0\)

The type of the expansion point is first determined. This is done on the homogeneous part of the ODE.

\[ y^{\prime \prime }+y^{\prime }+y = 0 \]

The following is summary of singularities for the above ode. Writing the ode as

\begin{align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end{align*}

Where

\begin{align*} p(x) &= 1\\ q(x) &= 1\\ \end{align*}

Table 2.120: Table \(p(x),q(x)\) singularites.

\(p(x)=1\)
singularity	type

\(q(x)=1\)
singularity	type

Combining everything together gives the following summary of singularities for the ode as

Regular singular points : \([]\)

Irregular singular points : \([\infty ]\)

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}y \left (x \right )+y \left (x \right )=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +1=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-3}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} y_{1}\left (x \right )+\mathit {C2} y_{2}\left (x \right )+y_{p}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\mathit {C2} \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+y_{p}\left (x \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (x \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (x \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (x \right )=-y_{1}\left (x \right ) \left (\int \frac {y_{2}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right )+y_{2}\left (x \right ) \left (\int \frac {y_{1}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right ), f \left (x \right )=\frac {1}{x}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \\ -\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} & -\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\frac {\sqrt {3}\, {\mathrm e}^{-x}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {2 \sqrt {3}\, {\mathrm e}^{-\frac {x}{2}} \left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{x}d x \right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{x}d x \right )\right )}{3} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {\left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {\left (1+\mathrm {I} \sqrt {3}\right ) x}{2}\right )-\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (\frac {\left (\mathrm {I} \sqrt {3}-1\right ) x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}} \sqrt {3}}{3} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\mathit {C2} \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\frac {\left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {\left (1+\mathrm {I} \sqrt {3}\right ) x}{2}\right )-\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (\frac {\left (\mathrm {I} \sqrt {3}-1\right ) x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}} \sqrt {3}}{3} \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   checking if the LODE has constant coefficients 
   <- constant coefficients successful 
<- solving first the homogeneous part of the ODE successful`
 

Maple dsolve solution

Solving time : 0.043 (sec)
Leaf size : maple_leaf_size

dsolve(diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = 1/x,y(x), 
       series,x=0)
 

\[ \text {No solution found} \]

Mathematica DSolve solution

Solving time : 0.065 (sec)
Leaf size : 152

AsymptoticDSolveValue[{D[y[x],{x,2}]+D[y[x],x]+y[x]==1/x,{}}, 
       y[x],{x,0,5}]
 

\[ y(x)\to c_2 x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right )+c_1 \left (\frac {x^3}{6}-\frac {x^2}{2}+1\right )+x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right ) \left (\frac {41 x^6}{4320}+\frac {x^5}{120}-\frac {x^4}{96}-\frac {x^3}{18}+x+\log (x)\right )+\left (\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \left (-\frac {x^6}{180}+\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^2}{4}-x\right ) \]