第10章 10.1

(1)

$y = Ae^x + Be^{-x}$ ①

$y' = Ae^x - Be^{-x}$ ②

$y'' = Ae^x + Be^{-x}$ ③

①③の右より $y - y'' = 0$ $y = y''$

(2)

$y = Ax + Bx^2$ ①

$y' = A + 2Bx$ ②

$y'' = 2B$ ③

③を②に代入して $y' = A + y''x$ $A = y' - y''x$ ④

③、④を①に代入して $y = (y' - y''x)x + \frac{1}{2}y''x^2$ $x^2y'' - 2xy' + 2y = 0$

(3)

$y^2 = Ax^2 + B$ ①

$2yy' = 2Ax$ $yy' = Ax$ ② $A = \frac{y}{x}y'$ ②’

②を①に代入して $y^2 - xyy' = B$ ③

②を微分して $y'^2 + yy'' = A$ ④

③、④を①に代入して $y^2 = (y'^2 + yy'')x^2 + y^2 - xyy'$

$y'^2x^2 + yy''x^2 - xyy' = 0$

(4)

$y = 4x + \frac{B}{x}$ ①

$y' = A - \frac{B}{x^2}$ ②

②を①に代入して $y = \left(y' + \frac{B}{x^2}\right)x + \frac{B}{x}$ $\frac{B}{x} = \frac{y}{2} - \frac{y'x}{2}$ ③

①から $xy = 4x^2 + B$ $y + xy' = 2Ax$ $y' + y' + xy'' = 2A$ $A = y' + \frac{xy''}{2}$ ④

③、④を①に代入して $y = \left(y' + \frac{xy''}{2}\right)x + \frac{y}{2} - \frac{y'x}{2}$

$x^2 y'' + xy' - y = 0$

(5)

$y = A\sin(Cx) + B\cos(Cx)$ ①

$y' = AC\cos(Cx) - BC\sin(Cx)$ ②

$y'' = -AC^2\sin(Cx) - BC^2\cos(Cx)$ ③

①、②から

$y'' = -C^2 y$

$y'' + C^2 y = 0$