A systematic approach to obtain the analytical solution for linear second order ordinary differential equations: part I

Abstract

The Leibniz integrating factor yields a reliable and direct way to solve linear first order ordinary differential equations. However, attempts to extend the technique to second or higher order differential equations resulted in integrating factors that are function of both the dependent and the independent variable and, thus, are the result of partial differential equations. Due to the importance of second and higher order ordinary differential equations in Applied Mathematics, Physics and Engineering, a simpler integrating factor, i.e., function of the independent variable only, gives important results to many problems that, today, rely on a multitude of different and more specific solution methods. Hence, this manuscript introduces the concept of generalized integrating factor for linear ordinary differential equations of order n. The procedure is used to address linear second order equations with varying and with constant coefficients. The solutions are analytically derived by means of nested convolutions and a close relation between linear and nonlinear differential equations is established. One interesting aspect of the proposed formulation is the fact that both the analytical homogeneous and the analytical particular solutions are obtained in separate. Analytical solutions for Bessel, Cauchy–Euler and the constant coefficients cases are provided and compared to examples in the literature and using numerical methods. Special attention is given to the constant coefficients case, since its application in mechanical and electrical engineering is widespread. Thus, a set of continuous excitation—periodic and polynomial—and discontinuous excitation functions—Dirac’s delta and Heaviside step function—are studied and their analytical solution are given. The analytical results were compared to each established method, like undetermined coefficients, variation of parameters and Laplace transform, showing the exactness and convenience of this generalization of the Leibniz integrating factor.

Appendices

Appendix A Families of Riccati equations with simple particular solutions using substitution

Although the Riccati sister equation is a nonlinear differential equation, for many cases finding a particular solution is straightforward. Two cases will be addressed: when the particular solution is a constant and when the particular solution is a polynomial of the independent variable. The focus will be on Eq. 32, nonetheless, the analysis holds true for the equation with $f_{2,1,1}$.

1.1 A.1 Constant particular solution

To Eq. 32 admit a constant particular solution, the following conditions must hold,

$$\begin{aligned} c-\dot{m}=a, \end{aligned}$$

(A1)

and,

$$\begin{aligned} km=b, \end{aligned}$$

(A2)

where a and b are constants. Thus, Eq. 32 is simplified to

$$\begin{aligned} f_{2,1,2}^2=af_{2,1,2}-b \end{aligned}$$

(A3)

whose particular solutions are the roots of the above quadratic algebraic equation. The positive root is chosen,

$$\begin{aligned} f_{2,1,2}=\frac{a+\sqrt{a^{2}-4b}}{2}. \end{aligned}$$

(A4)

1.2 A.2 Polynomial particular solution

Let $f_{2,1,2}$ and the differential equation coefficients be polynomials,

$$\begin{aligned}{} & {} f_{2,1,2}(t)=z_{0}+z_{1}t+z_{2}t^2+\cdots +z_{a}t^{a}, \end{aligned}$$

(A5)

$$\begin{aligned}{} & {} m(t)=m_{0}+m_{1}t+m_{2}t^2+\cdots +m_{b}t^{b}, \end{aligned}$$

(A6)

$$\begin{aligned}{} & {} c(t)=c_{0}+c_{1}t+c_{2}t^2+\cdots +c_{d}t^{d}, \end{aligned}$$

(A7)

$$\begin{aligned}{} & {} k(t)=k_{0}+k_{1}t+k_{2}t^2+\cdots +k_{g}t^{g}. \end{aligned}$$

(A8)

Consequentely the polynomial degree, D(p(t)), of each of these coefficients and particular solution is: $D\left( f_{2,1,2}\right) =a$, $D\left( m\right) =b$, $D\left( c\right) =d$ and $D\left( k\right) =g$. When applying these polynomials to Eq. 32, the degrees of the polynomials to be equal between the left hand side and the right hand side, the degrees must satisfy the following relations,

$$\begin{aligned}{} & {} a=b-1;\;\;d\le b-1;\;\;g\le b-2,\;\;\text {or} \end{aligned}$$

(A9)

$$\begin{aligned}{} & {} a=d;\;\;b\le d-1;\;\;g\le 2d-b,\;\;\text {or} \end{aligned}$$

(A10)

$$\begin{aligned}{} & {} a=\frac{b+g}{2};\;\;b\le g+2;\;\;d\le \frac{b+g}{2}. \end{aligned}$$

(A11)

Thus, applying these polynomials into Eq. 32 and separating the squares of each coefficient of the solution yields

$$\begin{aligned}{} & {} z_{0}^2+z_{1}^2 t^{2}+z_{2}^{2} t^{4}+\cdots +z_{a}^2 t^{2a}-z_{0}^2-z_{1}^2 t^{2}\nonumber \\{} & {} \qquad -z_{2}^2t^{4}-\cdots -z_{a}^2 t^{2a}+f_{2,1,2}^{2}\nonumber \\{} & {} \quad = \left( m_{0}+m_{1}t+m_{2}t^{2}+\cdots +m_{b}t^{b}\right) \nonumber \\{} & {} \qquad \left( z_{1}+2z_{2}t+3z_{3}t^{2} +\cdots +az_{a}t^{a-1}\right) \nonumber \\{} & {} \qquad + \left( c_{0}-m_{1}+t\left( c_{1}+2m_{2}\right) +t^{2}\left( c_{2}+3m_{3}\right) +\cdots \right) \nonumber \\{} & {} \qquad \left( z_{0}+z_{1}t+z_{2}t^2+\cdots +z_{a}t^{a}\right) \nonumber \\{} & {} \qquad - -m_{0}k_{0}-t\left( m_{0}k_{1}+m_{1}k_{0}\right) \nonumber \\{} & {} \quad \quad -t^{2}\left( m_{0}k_{2}+m_{1}k_{1}+m_{2}k_{0}\right) -\cdots . \end{aligned}$$

(A12)

The even powers of t on the left hand side can be matched to the equivalent terms on the right hand side,

$$\begin{aligned}{} & {} z_{0}^2+z_{1}^2 t^{2}+z_{2}^{2} t^{4}+\cdots +z_{a}^2 t^{2a}-z_{0}^2-z_{1}^2 t^{2}\nonumber \\{} & {} -z_{2}^2t^{4}-\cdots -z_{a}^2 t^{2a}+f_{2,1,2}^{2}\nonumber \\{} & {} \quad =m_{2}z_{1}t^2+2m_{3}z_{2}t^4+\cdots +a m_{a}z_{b}t^{a+b-1}\nonumber \\{} & {} \qquad -m_{2}z_{1}t^2-2m_{3}z_{2}t^4-\cdots -a m_{a}z_{b}t^{a+b-1}\nonumber \\{} & {} \qquad + m\dot{f}_{2,1,2}+z_{0}\left( c_{0}-m_{1}\right) +z_{1}t^{2}\left( c_{1}-2m_{2}\right) \nonumber \\{} & {} \qquad +z_{2}t^{4}\left( c_{2}-3m_{3}\right) +\cdots -z_{0}\left( c_{0}-m_{1}\right) \nonumber \\{} & {} \qquad -z_{1}t^{2}\left( c_{1}-2m_{2}\right) -z_{2}t^{4}\left( c_{2}-3m_{3}\right) -\cdots \nonumber \\{} & {} \qquad +\left( c-\dot{m}\right) f_{2,1,2}-m_{0}d_{0}\nonumber \\{} & {} \qquad - t^{2}\left( m_{0}d_{2}+m_{1}d_{1}+m_{2}d_{0}\right) \nonumber \\{} & {} \qquad -t^{4}\left( m_{0}d_{4}+m_{1}d_{3}+m_{2}d_{2} +m_{3}d_{1}+m_{4}d_{0}\right) -\cdots \nonumber \\{} & {} \qquad + m_{0}d_{0}+t^{2}\left( m_{0}d_{2}+m_{1}d_{1}+m_{2}d_{0}\right) \nonumber \\{} & {} \qquad +t^{4}\left( m_{0}d_{4}+m_{1}d_{3}+m_{2}d_{2}+m_{3}d_{1}+m_{4}d_{0}\right) \nonumber \\{} & {} \qquad +\cdots -md. \end{aligned}$$

(A13)

where from, one can derive the following independent relations,

$$\begin{aligned}{} & {} z_{0}^{2}=z_{0}\left( c_{0}-m_{1}\right) -m_{0}d_{0},\nonumber \\{} & {} \quad z_{1}^{2}=z_{1}\left( c_{1}-m_{2}\right) -m_{0}d_{2}-m_{1}d_{1}-m_{2}d_{0},\nonumber \\{} & {} \quad z_{2}^{2}=z_{2}\left( c_{2}-m_{3}\right) -m_{0}d_{4}\nonumber \\{} & {} \quad -m_{1}d_{3}-m_{2}d_{2}-m_{3}d_{1}-m_{4}d_{0},\nonumber \\{} & {} \quad \vdots , \end{aligned}$$

(A14)

which form a set of independent second order algebraic equations. When the z coefficients are applied into Eq. 32, all the even powers in the right hand side of the equation will cancel out with the even powers in the left hand side. Thus, the coefficients given in Eq. A14 to be solution of the Riccati sister equation, the coefficients of m, c and k must satisfy conditions given by the odd powers, which form a set of a equations.

Appendix B Families of Riccati equations with simple particular solutions using integration conditions

The given Riccati equation can be divided in 4 terms,

$$\begin{aligned} \underbrace{f_{2,1,2}^{2}}_{\phi _{1}}=\underbrace{m\dot{f}_{2,1,2}}_{\phi _{2}}+\underbrace{\left( c-\dot{m}\right) f_{2,1,2}}_{\phi _{3}}-\underbrace{md}_{\phi _{4}}. \end{aligned}$$

(B15)

Therefore, one can solve this equation piecewise. E.g.,

• $\phi _{1}=\phi _{2}$ and $\phi _{3}=-\phi _{4}$

  $$\begin{aligned} f_{2,1,2}^{2}=m\dot{f}_{2,1,2}\implies f_{2,1,2}=\frac{-1}{a+\int {\frac{1}{m} \text {d}t}}, \end{aligned}$$

  (B16)

  in which, a is a constant, and

  $$\begin{aligned} \left( c-\dot{m}\right) f_{2,1,2}=md \implies f_{2,1,2}=\frac{md}{c-\dot{m}}, \end{aligned}$$

  (B17)

  comparing both equations, one finds a condition for d,

  $$\begin{aligned} d=\frac{\dot{m}-c}{m\left( a+\int {\frac{1}{m} \text {d}t}\right) }. \end{aligned}$$

  (B18)

• $\phi _{1}=\phi _{4}$ and $\phi _{2}=-\phi _{3}$

  $$\begin{aligned}{} & {} m\dot{f}_{2,1,2}+\left( \dot{m}-c\right) f_{2,1,2}=0\nonumber \\{} & {} \quad \implies f_{2,1,2}=\frac{a\exp \left( \int {\frac{c}{m}\text {d}t}\right) }{m}, \end{aligned}$$

  (B19)

  and

  $$\begin{aligned} f_{2,1,2}^{2}=-md\implies f_{2,1,2}=\sqrt{-md}, \end{aligned}$$

  (B20)

  hence, the condition for d is

  $$\begin{aligned} d=\frac{a\exp \left( 2\int {\frac{c}{m}\text {d}t}\right) }{m^{3}}. \end{aligned}$$

  (B21)

  The negative sign and the square of constant a were omitted, since a can be any complex number.

• $\phi _{1}=\phi _{3}$ and $\phi _{2}=\phi _{4}$

  $$\begin{aligned} f_{2,1,2}^{2}=\left( c-\dot{m}\right) f_{2,1,2}\implies f_{2,1,2}=c-\dot{m}, \end{aligned}$$

  (B22)

  and,

  $$\begin{aligned} m\dot{f}_{2,1,2}=md\implies d=\dot{f}_{2,1,2}, \end{aligned}$$

  (B23)

  thus,

  $$\begin{aligned} d=\dot{c}-\ddot{m}. \end{aligned}$$

  (B24)

Integration conditions Eqs. B18 and B21 can be more useful, since the constant a in them generate an infinite number of possible functions d.

Appendix C Convolution over Dirac’s delta distribution

The convolution of a function over the Dirac’s delta is usually defined as [40]

$$\begin{aligned} \int _{-\infty }^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t=f\left( t_{k}\right) . \end{aligned}$$

(C25)

The integration limits can be split as

$$\begin{aligned}{} & {} \int _{-\infty }^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t=\int _{-\infty }^{0} f(t)\delta \left( t-t_{k}\right) \textrm{d}t\nonumber \\{} & {} \quad +\int _{0}^{t} f(t)\delta \left( t-t_{k}\right) \textrm{d}t\nonumber \\{} & {} \quad +\int _{t}^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t, \end{aligned}$$

(C26)

for $t_{k}$ strictly positive, the integral from $-\infty$ to 0 is 0 by definition, thus, the integral from 0 to t can be rewritten as

$$\begin{aligned}{} & {} \int _{0}^{t} f(t)\delta \left( t-t_{k}\right) \textrm{d}t=\int _{-\infty }^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t\nonumber \\{} & {} \quad -\int _{t}^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t. \end{aligned}$$

(C27)

The filter or sifting property of the delta of Dirac is due to the shape of this distribution, i.e., it is null everywhere except in its discontinuity, thus, the function that multiplies the Dirac’s delta is constant at this point, for the discontinuity of the delta distribution is infinitely close to the $t_{k}$ point. Therefore, the value of the function can be taken out of the integral and the definition of the Dirac’s delta is used to show that

$$\begin{aligned}{} & {} \int _{-\infty }^{\infty } f(t)\delta \left( t-t_{k}\right) \textrm{d}t=\int _{t_{k}-\tau }^{t_{k}+\tau } f(t)\delta \left( t-t_{k}\right) \textrm{d}t\nonumber \\{} & {} \quad =f(t_{k})\int _{t_{k}-\tau }^{t_{k}+\tau } \delta \left( t-t_{k}\right) \textrm{d}t\nonumber \\{} & {} \quad =f(t_{k}), \end{aligned}$$

(C28)

hence, Eq. C27 can be rewritten to

$$\begin{aligned}{} & {} \int _{0}^{t} f(t)\delta \left( t-t_{k}\right) \textrm{d}t=f(t_{k})\nonumber \\{} & {} \quad -f(t_{k}) {\left\{ \begin{array}{ll} 0,&{} \text { } t \ge t_{k}\\ 1,&{} \text { } t < t_{k} \end{array}\right. }=f(t_{k})\mathcal {H}(t-t_{k}). \end{aligned}$$

(C29)