2.1.25 Problem 25

Internal problem ID [9096]
Book : Second order enumerated odes
Section : section 1
Problem number : 25
Date solved : Sunday, March 30, 2025 at 02:06:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solved as second order linear constant coeff ode

Time used: 0.159 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime \prime }+y^{\prime }+y& =x\end{array}$$

This is second order non-homogeneous ODE. In standard form the ODE is

$$A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=f(x)$$

Where $A=1,B=1,C=1,f(x)=x$. Let the solution be

$$y=y_h+y_p$$

Where $y_h$ is the solution to the homogeneous ODE $A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=0$, and $y_p$ is a particular solution to the non-homogeneous ODE $A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=f(x)$. $y_h$ is the solution to

$$y^{\prime \prime }+y^{\prime }+y=0$$

This is second order with constant coefficients homogeneous ODE. In standard form the ODE is

$$A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=0$$

Where in the above $A=1,B=1,C=1$. Let the solution be $y=e^{\lambda x}$. Substituting this into the ODE gives

$$\begin{array}{}\text{(1)}& {\lambda }^2{\mathrm{e}}^{x\lambda }+\lambda {\mathrm{e}}^{x\lambda }+{\mathrm{e}}^{x\lambda }=0\end{array}$$

Since exponential function is never zero, then dividing Eq(2) throughout by $e^{\lambda x}$ gives

$$\begin{array}{}\text{(2)}& {\lambda }^2+\lambda +1=0\end{array}$$

Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula

$${\lambda }_{1,2}=\frac{-B}{2A}\pm \frac{1}{2A}\sqrt{B^2-4AC}$$

Substituting $A=1,B=1,C=1$ into the above gives

$$\begin{array}{rl}{\lambda }_{1,2}& =\frac{-1}{(2)\left(1\right)}\pm \frac{1}{(2)\left(1\right)}\sqrt{1^2-(4)\left(1\right)\left(1\right)}\\ & =-\frac{1}{2}\pm \frac{i\sqrt{3}}{2}\end{array}$$

Hence

$$\begin{array}{rl}{\lambda }_1& =-\frac{1}{2}+\frac{i\sqrt{3}}{2}\\ {\lambda }_2& =-\frac{1}{2}-\frac{i\sqrt{3}}{2}\end{array}$$

Which simplifies to

$$\begin{array}{rl}{\lambda }_1& =-\frac{1}{2}+\frac{i\sqrt{3}}{2}\\ {\lambda }_2& =-\frac{1}{2}-\frac{i\sqrt{3}}{2}\end{array}$$

Since roots are complex conjugate of each others, then let the roots be

$${\lambda }_{1,2}=\alpha \pm i\beta$$

Where $\alpha =-\frac{1}{2}$ and $\beta =\frac{\sqrt{3}}{2}$. Therefore the final solution, when using Euler relation, can be written as

$$y=e^{\alpha x}(c_1\cos (\beta x)+c_2\sin (\beta x))$$

Which becomes

$$y=e^{-\frac{x}{2}}(c_1\cos \left(\frac{\sqrt{3}x}{2}\right)+c_2\sin \left(\frac{\sqrt{3}x}{2}\right))$$

Therefore the homogeneous solution $y_h$ is

$$y_h={\mathrm{e}}^{-\frac{x}{2}}(c_1\cos \left(\frac{\sqrt{3}x}{2}\right)+c_2\sin \left(\frac{\sqrt{3}x}{2}\right))$$

The particular solution is now found using the method of undetermined coefficients. Looking at the RHS of the ode, which is

$$x$$

Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is

$$[\{1,x\}]$$

While the set of the basis functions for the homogeneous solution found earlier is

$$\{{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right),{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)\}$$

Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set.

$$y_p=A_{2}x+A_1$$

The unknowns $\{A_1,A_2\}$ are found by substituting the above trial solution $y_p$ into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives

$$A_{2}x+A_1+A_2=x$$

Solving for the unknowns by comparing coefficients results in

$$[A_1=-1,A_2=1]$$

Substituting the above back in the above trial solution $y_p$, gives the particular solution

$$y_p=x-1$$

Therefore the general solution is

$$\begin{array}{rl}y& =y_h+y_p\\ & =\left({\mathrm{e}}^{-\frac{x}{2}}(c_1\cos \left(\frac{\sqrt{3}x}{2}\right)+c_2\sin \left(\frac{\sqrt{3}x}{2}\right))\right)+(x-1)\end{array}$$

Will add steps showing solving for IC soon.

Summary of solutions found

$$\begin{array}{rl}y& =x-1+{\mathrm{e}}^{-\frac{x}{2}}(c_1\cos \left(\frac{\sqrt{3}x}{2}\right)+c_2\sin \left(\frac{\sqrt{3}x}{2}\right))\end{array}$$

Figure 2.51: Slope field $y^{\prime \prime }+y^{\prime }+y=x$

Solved as second order ode using Kovacic algorithm

Time used: 0.297 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime \prime }+y^{\prime }+y& =x\end{array}$$

Writing the ode as

$$\begin{array}{}\text{(1)}& y^{\prime \prime }+y^{\prime }+y& =0\text{(2)}& A{y}^{\prime \prime }+B{y}^{\prime }+Cy& =0\end{array}$$

Comparing (1) and (2) shows that

$$\begin{array}{rl}A& =1\\ \text{(3)}& B& =1C& =1\end{array}$$

Applying the Liouville transformation on the dependent variable gives

$$\begin{array}{rl}z(x)& =y{e}^{\int \frac{B}{2A}dx}\end{array}$$

Then (2) becomes

$$\begin{array}{}\text{(4)}& z^{″}(x)=rz(x)\end{array}$$

Where $r$ is given by

$$\begin{array}{rl}\text{(5)}& r& =\frac{s}{t}& =\frac{2A{B}^{\prime }-2B{A}^{\prime }+B^2-4AC}{4A^2}\end{array}$$

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

$$\begin{array}{}\text{(6)}& r& =\frac{-3}{4}\end{array}$$

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =-3\\ t& =4\end{array}$$

Therefore eq. (4) becomes

$$\begin{array}{}\text{(7)}& z^{″}(x)& =-\frac{3z\left(x\right)}{4}\end{array}$$

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

$$\begin{array}{rl}y& =z\left(x\right)e^{-\int \frac{B}{2A}dx}\end{array}$$

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 $\mathrm{\infty }$. The following table summarizes these cases.

Case	Allowed pole order for $r$	Allowed value for $\mathcal{O}(\mathrm{\infty })$
1	$\{0,1,2,4,6,8,\cdots \}$	$\{\cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \}$
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	$\{1,2\}$	$\{2,3,4,5,6,7,\cdots \}$

Table 2.10: Necessary conditions for each Kovacic case

The order of $r$ at $\mathrm{\infty }$ is the degree of $t$ minus the degree of $s$. Therefore

$$\begin{array}{rl}O\left(\mathrm{\infty }\right)& =\text{deg}(t)-\text{deg}(s)\\ & =0-0\\ & =0\end{array}$$

There are no poles in $r$. Therefore the set of poles $\mathrm{\Gamma }$ is empty. Since there is no odd order pole larger than $2$ and the order at $\mathrm{\infty }$ is $0$ then the necessary conditions for case one are met. Therefore

$$\begin{array}{rl}L& =[1]\end{array}$$

Since $r=-\frac{3}{4}$ is not a function of $x$, then there is no need run Kovacic algorithm to obtain a solution for transformed ode $z^{″}=rz$ as one solution is

$$z_1(x)=\cos \left(\frac{\sqrt{3}x}{2}\right)$$

Using the above, the solution for the original ode can now be found. The first solution to the original ode in $y$ is found from

$$\begin{array}{rl}{y}_1& =z_1{e}^{\int -\frac{1}{2}\frac{B}{A}dx}\\ & =z_1{e}^{-\int \frac{1}{2}\frac{1}{1}dx}\\ & =z_1{e}^{-\frac{x}{2}}\\ & =z_1\left({\mathrm{e}}^{-\frac{x}{2}}\right)\end{array}$$

Which simplifies to

$$y_1={\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)$$

The second solution $y_2$ to the original ode is found using reduction of order

$$y_2=y_1\int \frac{e^{\int -\frac{B}{A}dx}}{y_1^2}dx$$

Substituting gives

$$\begin{array}{rl}{y}_2& =y_1\int \frac{e^{\int -\frac{1}{1}dx}}{{\left(y_1\right)}^2}dx\\ & =y_1\int \frac{e^{-x}}{{\left(y_1\right)}^2}dx\\ & =y_1\left(\frac{2\sqrt{3}\tan \left(\frac{\sqrt{3}x}{2}\right)}{3}\right)\end{array}$$

Therefore the solution is

$$\begin{array}{rl}y& =c_1{y}_1+c_2{y}_2\\ & =c_1({\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right))+c_2({\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)\left(\frac{2\sqrt{3}\tan \left(\frac{\sqrt{3}x}{2}\right)}{3}\right))\end{array}$$

This is second order nonhomogeneous ODE. Let the solution be

$$y=y_h+y_p$$

Where $y_h$ is the solution to the homogeneous ODE $A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=0$, and $y_p$ is a particular solution to the nonhomogeneous ODE $A{y}^{″}(x)+B{y}^{\prime }(x)+Cy(x)=f(x)$. $y_h$ is the solution to

$$y^{\prime \prime }+y^{\prime }+y=0$$

The homogeneous solution is found using the Kovacic algorithm which results in

$$y_h=c_1{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\frac{2c_2{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{3}$$

The particular solution is now found using the method of undetermined coefficients. Looking at the RHS of the ode, which is

$$x$$

Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is

$$[\{1,x\}]$$

While the set of the basis functions for the homogeneous solution found earlier is

$$\{{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right),\frac{2{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{3}\}$$

Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set.

$$y_p=A_{2}x+A_1$$

The unknowns $\{A_1,A_2\}$ are found by substituting the above trial solution $y_p$ into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives

$$A_{2}x+A_1+A_2=x$$

Solving for the unknowns by comparing coefficients results in

$$[A_1=-1,A_2=1]$$

Substituting the above back in the above trial solution $y_p$, gives the particular solution

$$y_p=x-1$$

Therefore the general solution is

$$\begin{array}{rl}y& =y_h+y_p\\ & =(c_1{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\frac{2c_2{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{3})+(x-1)\end{array}$$

Will add steps showing solving for IC soon.

Summary of solutions found

$$\begin{array}{rl}y& =x-1+c_1{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\frac{2c_2{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{3}\end{array}$$

Figure 2.52: Slope field $y^{\prime \prime }+y^{\prime }+y=x$

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = x; 
dsolve(ode,y(x), singsol=all);
 

$$y={\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)c_2+{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)c_1+x-1$$

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 step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)+\frac{d}{dx}y\left(x\right)+y\left(x\right)=x\\ \bullet & & \text{Highest derivative means the order of the ODE is}2\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Characteristic polynomial of homogeneous ODE}\\ & & r^2+r+1=0\\ \bullet & & \text{Use quadratic formula to solve for}r\\ & & r=\frac{(-1)\pm \left(\sqrt{-3}\right)}{2}\\ \bullet & & \text{Roots of the characteristic polynomial}\\ & & r=(-\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2},-\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2})\\ \bullet & & \text{1st solution of the homogeneous ODE}\\ & & y_1\left(x\right)={\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)\\ \bullet & & \text{2nd solution of the homogeneous ODE}\\ & & y_2\left(x\right)={\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)\\ \bullet & & \text{General solution of the ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{y}_1\left(x\right)+\mathit{C}\mathit{2}{y}_2\left(x\right)+y_p\left(x\right)\\ \bullet & & \text{Substitute in solutions of the homogeneous ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)+y_p\left(x\right)\\ ◻& & \text{Find a particular solution}{y}_p\left(x\right)\text{of the ODE}\\ & \circ & \text{Use variation of parameters to find}{y}_p\text{here}f\left(x\right)\text{is the forcing function}\\ & & [y_p\left(x\right)=-y_1\left(x\right)\int \frac{y_2\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx+y_2\left(x\right)\int \frac{y_1\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx,f\left(x\right)=x]\\ & \circ & \text{Wronskian of solutions of the homogeneous equation}\\ & & W(y_1\left(x\right),y_2\left(x\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}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{2}& -\frac{{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)}{2}+\frac{{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\cos \left(\frac{\sqrt{3}x}{2}\right)}{2}\end{array}\right]\\ & \circ & \text{Compute Wronskian}\\ & & W(y_1\left(x\right),y_2\left(x\right))=\frac{\sqrt{3}{\mathrm{e}}^{-x}}{2}\\ & \circ & \text{Substitute functions into equation for}{y}_p\left(x\right)\\ & & y_p\left(x\right)=-\frac{2\sqrt{3}{\mathrm{e}}^{-\frac{x}{2}}(\int x{\mathrm{e}}^{\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)dx\cos \left(\frac{\sqrt{3}x}{2}\right)-\int x{\mathrm{e}}^{\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)dx\sin \left(\frac{\sqrt{3}x}{2}\right))}{3}\\ & \circ & \text{Compute integrals}\\ & & y_p\left(x\right)=-1+x\\ \bullet & & \text{Substitute particular solution into general solution to ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)-1+x\end{array}$$

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 50

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

$$y(x)\to x+c_2{e}^{-x/2}\cos \left(\frac{\sqrt{3}x}{2}\right)+c_1{e}^{-x/2}\sin \left(\frac{\sqrt{3}x}{2}\right)-1$$

✓ Sympy. Time used: 0.159 (sec). Leaf size: 34

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

$$y\left(x\right)=x+(C_1\sin \left(\frac{\sqrt{3}x}{2}\right)+C_2\cos \left(\frac{\sqrt{3}x}{2}\right))e^{-\frac{x}{2}}-1$$