Internal problem ID [11495]
Internal file name [OUTPUT/10478_Thursday_May_18_2023_04_20_39_AM_11046977/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises
page 125
Problem number: 1.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "reduction_of_order", "exact linear second order ode", "second_order_integrable_as_is"
Maple gives the following as the ode type
[[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime } t +x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{-\frac {t^{2}}{2}} \end {align*}
Given one basis solution \(x_{1}\left (t \right )\), then the second basis solution is given by \[ x_{2}\left (t \right ) = x_{1} \left (\int \frac {{\mathrm e}^{-\left (\int p d t \right )}}{x_{1}^{2}}d t \right ) \] Where \(p(x)\) is the coefficient of \(x^{\prime }\) when the ode is written in the normal form \[ x^{\prime \prime }+p \left (t \right ) x^{\prime }+q \left (t \right ) x = f \left (t \right ) \] Looking at the ode to solve shows that \[ p \left (t \right ) = t \] Therefore \begin{align*} x_{2}\left (t \right ) &= {\mathrm e}^{-\frac {t^{2}}{2}} \left (\int {\mathrm e}^{-\left (\int t d t \right )} {\mathrm e}^{t^{2}}d t \right ) \\ x_{2}\left (t \right ) &= {\mathrm e}^{-\frac {t^{2}}{2}} \int \frac {{\mathrm e}^{-\frac {t^{2}}{2}}}{{\mathrm e}^{-t^{2}}} , dt \\ x_{2}\left (t \right ) &= {\mathrm e}^{-\frac {t^{2}}{2}} \left (\int {\mathrm e}^{\frac {t^{2}}{2}}d t \right ) \\ x_{2}\left (t \right ) &= -\frac {i {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \\ \end{align*} Hence the solution is \begin{align*} x &= c_{1} x_{1}\left (t \right )+c_{2} x_{2}\left (t \right ) \\ &= {\mathrm e}^{-\frac {t^{2}}{2}} c_{1} -\frac {i c_{2} {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} x &= {\mathrm e}^{-\frac {t^{2}}{2}} c_{1} -\frac {i c_{2} {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \\ \end{align*}
Verification of solutions
\[ x = {\mathrm e}^{-\frac {t^{2}}{2}} c_{1} -\frac {i c_{2} {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x^{\prime }+x^{\prime } t +x=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d t}x^{\prime } \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} x \\ {} & {} & x=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} t^{k} \\ \square & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} t \cdot x^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & t \cdot x^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} k \,t^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d t}x^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x^{\prime }=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) t^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \frac {d}{d t}x^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) t^{k} \\ & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & {} & \moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +2} \left (k +2\right ) \left (k +1\right )+a_{k} \left (k +1\right )\right ) t^{k}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k +1\right ) \left (a_{k +2} \left (k +2\right )+a_{k}\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [x=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} t^{k}, a_{k +2}=-\frac {a_{k}}{k +2}\right ] \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve([diff(x(t),t$2)+t*diff(x(t),t)+x(t)=0,exp(-t^2/2)],singsol=all)
\[ x \left (t \right ) = \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 41
DSolve[x''[t]+t*x'[t]+x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{2} e^{-\frac {t^2}{2}} \left (\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {t}{\sqrt {2}}\right )+2 c_2\right ) \]