@article{oai:ynu.repo.nii.ac.jp:00006621,
 author = {Kaizuka, Tetsu},
 issue = {2},
 journal = {Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学},
 month = {},
 note = {application/pdf, In this paper we will discuss methods for the numerical solution of a single first-order ordinary differential equation ＄￥frac{dy}{dx}=f(x, y)￥equiv￥frac{Q(x,y)}{P(x,y)}＄ with the initial condition ＄y(a)=b＄ . To seek the solution, we often use Runge-Kutta method provided that ＄f(x, y)＄ is continuous and satisfies Lipschitz condition ＄|f(x, y_{1})-f(x, y_{2})|￥leqq M|y_{1}-y_{2}|＄ . If ＄P(x_{0}, y_{0})=0＄ and ＄Q(x_{0}, y_{0})￥neq 0＄ , then ＄ dyldx=￥infty＄ , therefore we can not use Runge-Kutta method in the neighbourhood of ＄(x_{0}, y_{0})＄ . Nevertheless, there often exists the solution. (ex. 1) In this case, putting ＄g(x, y)=￥frac{P(x,y)}{Q(x,y)}＄ ＄g(x, y)＄ often satisfies Lipschitz condition ＄|g(x_{1}, y)-g(x_{2}, y)|￥leqq M^{￥prime}|x_{1}-x_{2}|＄ in the neighbourhood of ＄(x_{0}, y_{0})＄ . Therefore, when we use Runge-kutta method provided with ＄P(x, y)^{2}+Q(x, y)^{2}￥neq 0＄ , if ＄|P(x, y)|￥geqq|Q(x, y)|＄ , we treat ＄y＄ as the function of the independent variable ＄x＄ , if ＄|P(x, y)|<|Q(x, y)|＄ , we treat ＄x＄ as the function of the independent variable ＄y＄ , that is, if ＄|￥frac{dy}{dx}|￥leqq 1＄ , then ￥frac{dy}{dx}=￥frac{Q(X_{1}y)}{P(X_{1}y)}＄ , and if ＄|￥frac{dy}{dx}|>1＄ , then ＄￥frac{dx}{dy}=￥frac{P(x,y)}{Q(X_{1}y)}＄ , then, as ＄|dy/dx|＄ and ＄|dx/dy|＄ are both smaller than 1, the error of the increment will be smaller. If ＄P(x_{0}, y_{0})^{2}+Q(x_{0}, y_{0})^{2}=0＄ , we must stop pursuing the solution at ＄(x_{0}, y_{0})＄ as ＄(x_{0}, y_{0})＄ is a singular point. (ex. ＄6￥sim ex＄ . ＄11＄ ) Furthermore, if we can find proper continuations between dyldx and dxldy, we can even pursue closed curves. (ex. 1)},
 pages = {183--189},
 title = {A REMARK ON THE NUMERICAL SOLUTION OF A SINGLE FIRST-ORDER ORDINARY DIFFERENTIAL EQUATION},
 volume = {25},
 year = {1977}
}