2.1.787 Problem 809

Internal problem ID [11247]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 809
Date solved : Monday, December 08, 2025 at 10:33:19 PM
CAS classification : [[_2nd_order, _missing_x]]

2.1.787.1 Solved as second order ode using Kovacic algorithm

0.050 (sec)

\begin{align*} u^{\prime \prime }+2 u^{\prime }+u&=0 \\ \end{align*}

Entering kovacic solverWriting the ode as

\begin{align*} u^{\prime \prime }+2 u^{\prime }+u &= 0 \tag {1} \\ A u^{\prime \prime } + B u^{\prime } + C u &= 0 \tag {2} \end{align*}

Comparing (1) and (2) shows that

\begin{align*} A &= 1 \\ B &= 2\tag {3} \\ C &= 1 \end{align*}

Applying the Liouville transformation on the dependent variable gives

\begin{align*} z(x) &= u e^{\int \frac {B}{2 A} \,dx} \end{align*}

Then (2) becomes

\begin{align*} z''(x) = r z(x)\tag {4} \end{align*}

Where \(r\) is given by

\begin{align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end{align*}

Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives

\begin{align*} r &= \frac {0}{1}\tag {6} \end{align*}

Comparing the above to (5) shows that

\begin{align*} s &= 0\\ t &= 1 \end{align*}

Therefore eq. (4) becomes

\begin{align*} z''(x) &= 0 \tag {7} \end{align*}

Equation (7) is now solved. After finding \(z(x)\) then \(u\) is found using the inverse transformation

\begin{align*} u &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end{align*}

The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3 cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table summarizes these cases.

Case	Allowed pole order for \(r\)	Allowed value for \(\mathcal {O}(\infty )\)
1	\(\left \{ 0,1,2,4,6,8,\cdots \right \} \)	\(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \)
2	Need to have at least one pole that is either order \(2\) or odd order greater than \(2\). Any other pole order is allowed as long as the above condition is satisfied. Hence the following set of pole orders are all allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\).	no condition
3	\(\left \{ 1,2\right \} \)	\(\left \{ 2,3,4,5,6,7,\cdots \right \} \)

Table 2.787: Necessary conditions for each Kovacic case

The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore

\begin{align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 0 - -\infty \\ &= \infty \end{align*}

There are no poles in \(r\). Therefore the set of poles \(\Gamma \) is empty. Since there is no odd order pole larger than \(2\) and the order at \(\infty \) is \(infinity\) then the necessary conditions for case one are met. Therefore

\begin{align*} L &= [1] \end{align*}

Since \(r = 0\) is not a function of \(x\), then there is no need run Kovacic algorithm to obtain a solution for transformed ode \(z''=r z\) as one solution is

\[ z_1(x) = 1 \]

Using the above, the solution for the original ode can now be found. The first solution to the original ode in \(u\) is found from

\begin{align*} u_1 &= z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dx} \\ &= z_1 e^{ -\int \frac {1}{2} \frac {2}{1} \,dx} \\ &= z_1 e^{-x} \\ &= z_1 \left ({\mathrm e}^{-x}\right ) \\ \end{align*}

Which simplifies to

\[ u_1 = {\mathrm e}^{-x} \]

The second solution \(u_2\) to the original ode is found using reduction of order

\[ u_2 = u_1 \int \frac { e^{\int -\frac {B}{A} \,dx}}{u_1^2} \,dx \]

Substituting gives

\begin{align*} u_2 &= u_1 \int \frac { e^{\int -\frac {2}{1} \,dx}}{\left (u_1\right )^2} \,dx \\ &= u_1 \int \frac { e^{-2 x}}{\left (u_1\right )^2} \,dx \\ &= u_1 \left (x\right ) \\ \end{align*}

Therefore the solution is

\begin{align*} u &= c_1 u_1 + c_2 u_2 \\ &= c_1 \left ({\mathrm e}^{-x}\right ) + c_2 \left ({\mathrm e}^{-x}\left (x\right )\right ) \\ \end{align*}

Figure 2.3: Slope field \(u^{\prime \prime }+2 u^{\prime }+u = 0\)

2.1.787.2 ✓ Maple. Time used: 0.001 (sec). Leaf size: 14

ode:=diff(diff(u(x),x),x)+2*diff(u(x),x)+u(x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = {\mathrm e}^{-x} \left (c_2 x +c_1 \right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}u \left (x \right )+2 \frac {d}{d x}u \left (x \right )+u \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}u \left (x \right ) \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}+2 r +1=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \left (r +1\right )^{2}=0 \\ \bullet & {} & \textrm {Root of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =-1 \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & u_{1}\left (x \right )={\mathrm e}^{-x} \\ \bullet & {} & \textrm {Repeated root, multiply}\hspace {3pt} u_{1}\left (x \right )\hspace {3pt}\textrm {by}\hspace {3pt} x \hspace {3pt}\textrm {to ensure linear independence}\hspace {3pt} \\ {} & {} & u_{2}\left (x \right )=x \,{\mathrm e}^{-x} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & u \left (x \right )=\mathit {C1} u_{1}\left (x \right )+\mathit {C2} u_{2}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & u \left (x \right )=\mathit {C1} \,{\mathrm e}^{-x}+\mathit {C2} x \,{\mathrm e}^{-x} \end {array} \]

2.1.787.3 ✓ Mathematica. Time used: 0.01 (sec). Leaf size: 18

ode=D[u[x],{x,2}]+2*D[u[x],x]+u[x]==0; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to e^{-x} (c_2 x+c_1) \end{align*}

2.1.787.4 ✓ Sympy. Time used: 0.076 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
u = Function("u") 
ode = Eq(u(x) + 2*Derivative(u(x), x) + Derivative(u(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- x} \]