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A NOTE ON THE THEORY OF THE RICCATI EQUATION
E. I. GRUDO
In [1] it is shown that the solution of the equation
\[ y'=y^2+f(x), \tag{1} \]
where \(f(x)\) is a function holomorphic in the domain \(|x|<\rho\), with the initial condition \(y(0)=0\), is holomorphic at least in the disk
\[ |x|<R_1=\frac{1}{\sqrt{M+\frac{1}{R^2}}}, \tag{2} \]
where \(M=\max_{|x|\le R}|f(x)|,\quad R<\rho\). Estimate (2) for the radius of convergence is better than the estimate given by Cauchy’s theorem [2]. To derive estimate (2), an estimate of the Maclaurin coefficients of the indicated holomorphic solution of equation (1) is carried out by means of mathematical induction. However, it is very simple to obtain for the radius of convergence \(R_1\) an estimate better than (2).
For this purpose let us consider the equation
\[ y'=y^2+\frac{M}{\left(1-\frac{x}{R}\right)^2}. \tag{3} \]
The solution of this equation with the initial condition \(y(0)=0\) obviously majorizes the solution of equation (1) with the same initial condition. But equation (3) is easily integrated in elementary functions.
Without loss of generality we shall distinguish two cases: 1) \(4R^2M=1\) and 2) \(4R^2M>1\), since the case \(4R^2M<1\) reduces to case 1) by taking in equation (3), instead of \(M\), the constant
\[ M_1=\frac{1}{4R^2}>M. \]
In case 1) the solution of equation (3) with the initial condition \(y(0)=0\) is
\[ y=\frac{\ln(R-x)-\ln R}{2(R-x)\,[\ln(R-x)-\ln R-2]}, \tag{4} \]
and in case 2)
\[ y= \frac{\lambda_1(R-x)^{\lambda_1-1} -\lambda_1 R^{\lambda_1-\lambda_2}(R-x)^{\lambda_2-1}} {(R-x)^{\lambda_1} -\frac{\lambda_1}{\lambda_2}R^{\lambda_1-\lambda_2}(R-x)^{\lambda_2}}, \tag{5} \]
where
\[ \lambda_1=\frac{1+i\sqrt{4R^2M-1}}{2}, \qquad \lambda_2=\overline{\lambda_1}. \]
Obviously, solution (4) of equation (3) is holomorphic for \(|x|<R\). The denominator of solution (5) vanishes for \(x=R\) and
\[ x= \begin{cases} \left[1-\exp\left(\dfrac{\operatorname{arc\,tg}\dfrac{2a}{1-a^2}}{a}\right)\right]R, & \text{if } 0<a<1,\\[1.2em] \left[1-\exp\left(\dfrac{\pi-\operatorname{arc\,tg}\dfrac{2a}{a^2-1}}{a}\right)\right]R, & \text{if } a>1, \end{cases} \tag{6} \]
where
\[ a=\sqrt{4R^2M-1}. \]
It is not difficult to verify that the modulus of the right-hand side of (6) for \(0<a<1\) is greater than \(R\), while for \(a>1\) it is a decreasing function of \(a\). Let us denote by \(a^*\) the root of the equation
\[ \pi-\operatorname{arc tg}\frac{2a}{a^2-1}=a\ln 2. \]
It is clear that the root \(a^*\) will be unique and \(a^*>1\). Thus, for \(1<a\leq a^*\) the right-hand side of (6) will have modulus not less than \(R\), and for \(a>a^*\) it will be less than \(R\).
Thus, if \(4R^2M\leq 1+a^{*2}\), then the solution of equation (1) with the initial condition \(y(0)=0\) is holomorphic in the disk
\[ |x|<R, \tag{7} \]
and if \(4R^2M>1+a^{*2}\), then this solution is holomorphic at least in the disk
\[ |x|< \left[ \exp\left(\frac{\pi-\operatorname{arc tg}\dfrac{2a}{a^2-1}}{a}\right)-1 \right]R, \qquad a=\sqrt{4R^2M-1}. \tag{8} \]
It is not difficult to see that the estimates of the radius of convergence (7) and (8) for the solution of equation (1) with the initial condition \(y(0)=0\) are better than the estimates (2) obtained in [1].
The coefficients of the Maclaurin expansion of functions (4) and (5) in the corresponding cases, for sufficiently large \(n\), evidently give better estimates for the coefficients of the Maclaurin expansion of the solution of equation (1) with the initial condition \(y(0)=0\) than the estimates obtained in [1].
The same method can also be applied to some equations of higher order.
References
- Todorov P. G. On the theory of the Riccati equation. Ukrainian Mathematical Journal, 18, No. 1, 137—139, 1966.
- Golubev V. V. Lectures on the Analytic Theory of Differential Equations. GITTL, Moscow—Leningrad, 1941.