An introduction to linear ordinary differential equations using the impulsive response method and factorization
This book develops the basic theory of \(n\)-th order linear differential equations of the form \[ y^{(n)}+a_1y^{(n-1)}+a_2y^{(n-2)}+\cdots+a_{n-1}y'+a_ny=f.\tag{1} \] In contrast to more traditional approaches (e.g., reduction to a first-order linear system, or the method of variation of parameters), the exposition is based on the method of factorization of linear differential operators. The book has two main parts; the first deals with the case when the coefficients \(a_1,\dots,a_n\) in (1) are constants and \(f\) is a continuous function. The basic idea is as follows: First, consider equation (1) together with the trivial initial conditions \[ y(x_0)=y'(x_0)=\cdots=y^{(n-1)}(x_0)=0.\tag{2} \] If \(\lambda_1,\dots,\lambda_n\) are the roots of the characteristic polynomial \[ \lambda^n+a_1\lambda^{n-1}+a_2\lambda^{n-2}+\cdots+a_{n-1}\lambda+a_n, \] then equation (1) can be rewritten as \[ (\frac{d}{dx}-\lambda_1)\cdots(\frac{d}{dx}-\lambda_n)y=f. \] By letting \(h=(\frac{d}{dx}-\lambda_n)y\), we get \[ (\frac{d}{dx}-\lambda_1)\cdots(\frac{d}{dx}-\lambda_{n-1})h=f,\quad h(x_0)=h'(x_0)=\cdots=h^{(n-2)}(x_0)=0, \] and \[ y'-\lambda_n y=h,\quad y(x_0)=0. \] Once we find \(h\), the solution of the first-order problem is \(y(x)=\int_{x_0}^x e^{\lambda_n(x-t)}h(t)dt\). Using the found \(h\), we apply the same procedure to the previous equation of order \(n-1\) and reduce it to a first-order equation and an equation of order \(n-2\). Proceeding further in the same way, one eventually finds that \[ y(x)=\int_{x_0}^x g_{\lambda_1,\ldots,\lambda_n}(x-s)f(s)ds,\tag{3} \] where \(g_{\lambda}(x)=e^{\lambda x}\) and \(g_{\lambda_1,\ldots,\lambda_k}=\int_{x_0}^x g_{\lambda_k}(x-t)g_{\lambda_1,\ldots,\lambda_{k-1}}(t)dt\) for \(k>1\). The function \(g=g_{\lambda_1,\ldots,\lambda_n}\) is called the impulse response (the reason becomes obvious in the context of distribution theory), and it is the unique solution of the homogeneous problem \[ y^{(n)}+a_1y^{(n-1)}+a_2y^{(n-2)}+\cdots+a_{n-1}y'+a_ny=0 \tag{4} \] satisfying the initial conditions \[ y(x_0)=y'(x_0)=\cdots=y^{(n-2)}(x_0)=0,\quad y^{(n-1)}(x_0)=1.\tag{5} \] Consequently, solutions of the homogeneous problem with arbitrary inital conditions at \(x=x_0\) can be obtained as linear combinations of \(g,g',\ldots,g^{(n-1)}\). By combining this result with (3), one obtains all solutions of equation (1) corresponding to arbitrary initial conditions. A fairly laborious calculation shows that if the characteristic polynomial has distinct roots \(\lambda_1,\ldots,\lambda_k\) with multiplicities \(m_1,\ldots,m_k\), then the impulse response can be written in the form \[ g(x)=G_1(x)e^{\lambda_1x}+\cdots+G_m(x)e^{\lambda_mx}, \] where each \(G_i\) is a polynomial of order \(m_i-1\). This leads to the well known general form of solutions to the homogeneous problem. Finally, the factorization and impulse response method can be used to obtain formulas for particular solutions of nonhomogeneous problems whose right-hand side has the special form \(f(x)=P(x)e^{\lambda_0x}\), where \(P\) is a complex polynomial (the result is known as the method of undetermined coefficients). The goal of the second part of the book is to generalize the previous results to the case when the coefficients \(a_1,\ldots,a_n\) in (1) are continuous functions. The main problem here is to find decomposition of the left-hand side of (1) of the form \[ (\frac{d}{dx}-\alpha_1(x))\cdots(\frac{d}{dx}-\alpha_n(x)).\tag{6} \] The proof that this is always possible is nontrivial and is omitted. It can be found in the literature; see [\textit{G. Mammana}, Math. Z. 33, 186--231 (1931; Zbl 0001.01504)] for the case of real-valued coefficients, and [\textit{R. Camporesi} and \textit{A. J. Di Scala}, Colloq. Math. 122, No. 2, 215--223 (2011; Zbl 1230.34011)]. By a similar procedure as in the constant-coefficient case, one finds that the solution of (1) corresponding to the initial conditions (2) is given by \[ y(x)=\int_{x_0}^x g_{\alpha_1,\ldots,\alpha_n}(x,t)f(t)dt,\tag{6} \] where \(g_{\alpha}(x,t)=e^{\int_t^x\alpha(s)ds}\) and \(g_{\alpha_1,\ldots,\alpha_k}(x,t)=\int_{t}^x g_{\alpha_k}(x,s)g_{\alpha_1,\ldots,\alpha_{k-1}}(s,t)ds\) for \(k>1\). As before, the impulse response \(g=g_{\alpha_1,\ldots,\alpha_n}\) provides a solution of the homogeneous problem (4) subject to the initial conditions (5), and each solution of (4) is a linear combination of \(g,\partial_t g,\ldots,\partial_t^{n-1}g\). One can also show that \[ g(x,t)=c_1(t)y_1(x)+\cdots+c_n(t)y_n(x), \] where \(y_1,\ldots,y_n\) is any fundamental system of (4) and \(c_1,\ldots,c_n\) can be calculated using certain Wronskians. Substitution into (6) then leads to the variation of constants formula. Unfortunately, there is no general method for finding the functions \(\alpha_1,\ldots,\alpha_n\) in (6). Consequently, except for special cases, there is no explicit formula for \(g\). For \(n=2\), the functions \(\alpha_1\), \(\alpha_2\) can be obtained by solving a Riccati equation. Alternatively, one can rewrite the \(n\)-th order homogeneous equation (4) as a system of first-order equations, and solve it using the method of successive approximations (this leads to the Peano-Baker series). In this way, one can obtain formulas that express \(g\) as infinite series of iterated integrals. Although the book does not contain new results, it seems to be the first systematic treatment of \(n\)-th order equations based on the factorization method. As it is to be expected, this approach has its advantages and disadvantages. The basic idea is very easy to understand, requires only modest theoretical background (e.g., no Jordan canonical form theory), and is easily demonstrated on examples. Along the way, the reader encounters other interesting topics, such as the convolution, Riccati equations, or disconjugacy. On the other hand, the proofs of some theoretical results seem to be more involved in comparison with more traditional approaches. In any case, the author convincingly demonstrates that the method of factorization represents a useful alternative approach. Both parts of the book begin with a separate treatment of first and second-order equations, and only then proceed to general equations of order \(n\). This structure, as well as numerous solved examples, significantly contribute to the overall readability of the book. Unsolved exercises are also included.
scientific article; zbMATH DE number 6675952