数学の基礎と公式-18
数学の基礎と公式
amazon kindle版を出版しました。
18. 微分方程式-3
$\displaystyle{ }$
Maximaで計算しています。
（同次形ほか）
${\it \%c},{\it \%k}_{2}\,{\it \%k}_{1}$は任意定数
※$\log x$の$x$の箇所には絶対値がつきます。$\log |x|$
(1) $\displaystyle{ \dfrac{dx}{dt}-2x-1=0 }$
$\displaystyle{ x={{2\,{\it \%c}\,e^{2\,t}-1}\over{2}} }$
(2) $\displaystyle{ \dfrac{dx}{dt}-2x-e^t=0 }$
$\displaystyle{ x=\left({\it \%c}-e^ {- t }\right)\,e^{2\,t} }$
(3) $\displaystyle{ \dfrac{dx}{dt}+x-t^2=0 }$
$\displaystyle{ x={\it \%c}\,e^ {- t }+t^2-2\,t+2 }$
(4) $\displaystyle{ \dfrac{dx}{dt}+2x-t^2-1=0 }$
$\displaystyle{ x={\it \%c}\,e^ {- 2\,t }+{{t^2}\over{2}}-{{t}\over{2 }}+{{3}\over{4}} }$
(5) $\displaystyle{ \dfrac{dx}{dt} t+x+t^2-1=0 }$
$\displaystyle{ x=-{{t^2}\over{3}}+{{{\it \%c}}\over{t}}+1 }$
(6) $\displaystyle{ \dfrac{dx}{dt} +2 x t-t e^{-t^2}=0 }$
$\displaystyle{ x=\left({{t^2}\over{2}}+{\it \%c}\right)\,e^ {- t^2 } }$
(7) $\displaystyle{ \dfrac{dx}{dt} + x e^{t}-5 e^t =0 }$
$\displaystyle{ x=e^ {- e^{t} }\,\left(5\,e^{e^{t}}+{\it \%c}\right) }$
(8) $\displaystyle{ \dfrac{dx}{dt} + x-t x^3 =0 }$
$\displaystyle{ x^2={{1}\over{{\it \%c}\,e^{2\,t}+t+{{1}\over{2}}}} }$
(9) $\displaystyle{ 2 \dfrac{dx}{dt} xt-x^2+t =0 }$
$\displaystyle{ {{t\,\log t+x^2}\over{t}}={\it \%c} }$
(10) $\displaystyle{ \dfrac{dx}{dt} x+\dfrac{t x^2}{2(1-t^2)}-t =0 }$
$\displaystyle{ {{\sqrt{t^2-1}\,\left(x^2-2\,t^2+2\right)}\over{2\,t^
2-2}}={\it \%c} }$
(11) $\displaystyle{ \dfrac{dx}{dt} x-\dfrac{t x^2+2t}{t^2-1} =0 }$
$\displaystyle{ {{\log \left(x^2+2\right)}\over{2}}={{\log \left(t+1 \right)+\log \left(t-1\right)+2\,{\it \%c}}\over{2}} }$
$\displaystyle{ \log \left(x^2+2\right)=\log \left(t^2-1 \right)+2\,{\it \%c} }$
$\displaystyle{ x^2={\it \%c}\left(t^2-1 \right)-2 }$
(11) $\displaystyle{ \dfrac{dx}{dt} t+x-x^2 \log(t) =0 }$
$\displaystyle{ x\left(t\right)={{1}\over{\log t+{\it \%c}\,t+1}} }$
(12) $\displaystyle{ \dfrac{dx}{dt} +x-x^2(\cos t-\sin t) =0 }$
$\displaystyle{ x={{1}\over{{\it \%c}\,e^{t}-\sin t}} }$
(13) $\displaystyle{ \dfrac{dx}{dt}t-2x+t =0 }$
$\displaystyle{ x={\it \%c}\,t^2+t }$
(14) $\displaystyle{ 2 \dfrac{dx}{dt}tx-x^2+t^2 =0 }$
$\displaystyle{ -{{t}\over{x^2+t^2}}={\it \%c} }$
$\displaystyle{ x^2+t^2={\it \%c}t }$
(15) $\displaystyle{ \dfrac{dx}{dt}+\dfrac{x}{t}-\log t =0 }$
$\displaystyle{ x={{t\,\log t}\over{2}}-{{t}\over{4}}+{{{\it \%c} }\over{t}} }$
(16) $\displaystyle{ \dfrac{dx}{dt}+\dfrac{x}{t}-x^2 t^2 =0 }$
$\displaystyle{ x={{1}\over{{\it \%c}\,t-{{t^3}\over{2}}}} }$
(17) $\displaystyle{ \dfrac{dx}{dt}+x+\dfrac{x^3 t^2}{2} =0 }$
$\displaystyle{ x={{1}\over{t\,\sqrt{t+{\it \%c}}}} }$
(18) $\displaystyle{ \dfrac{dx}{dt}-\dfrac{ x}{t-1}+x^2 =0 }$
$\displaystyle{ x={{2\,t-2}\over{t^2-2\,t-2\,{\it \%c}}} }$
(19) $\displaystyle{ \dfrac{dx}{dt}- x(1+x t) =0 }$
$\displaystyle{ x={{e^{t}}\over{{\it \%c}-\left(t-1\right)\,e^{t}}} }$
(20) $\displaystyle{ \dfrac{dx}{dt}- \dfrac{x}{t(t+1)(t+2)} =0 }$、$t=1$のとき$x(t)=1$
$\displaystyle{ x={{2\,\sqrt{t(t+2)}}\over{ \sqrt{3}\,\left(t+1\right)}} }$
(21) $\displaystyle{ \dfrac{dx}{dt}- 2 \sqrt{x} =0 }$、$t=0$のとき$x(t)=1$
$\displaystyle{ x=(t+1)^2 }$
(22) $\displaystyle{ \dfrac{dx}{dt}- \dfrac{x}{t}-t^2 =0 }$、$t=1$のとき$x(t)=4.5$
$\displaystyle{ x={{t^3+8t}\over{2 }} }$
(23) $\displaystyle{ \dfrac{dx}{dt} t-2x+t =0 }$、$t=1$のとき$x(t)=2$
$\displaystyle{ x=t^2+t }$
maximaでの記述
物理の物体の自由落下に関する問題で,上向きを正とした場合、
(1)$\displaystyle{ \dfrac{d^2 x}{d t^2}=-g }$
・ode2を使う場合
ode2( diff(x(t), t, 2) = -g, x(t), t );
2
g t
x(t) = (- ----) + %k2 t + %k1
2
$\displaystyle{ x\left(t\right)=-{{g\,t^2}\over{2}}+{\it \%k}_{2}\,t+{\it \%k}_{1}}$
${\it \%k}_{2}\,{\it \%k}_{1}$は任意定数
・desolveを使う場合
desolve( diff(x(t), t, 2) = -g, x(t) );
$\displaystyle{x\left(t\right)=t\,\left(\left.{{d}\over{d\,t}}\,x\left(t\right) \right|_{t=0}\right)-{{g\,t^2}\over{2}}+x\left(0\right)}$
$\displaystyle{x'(0)=\left(\left.{{d}\over{d\,t}}\,x\left(t\right) \right|_{t=0}\right)}$のことです。
(2)初期値$t=0$のとき$x=0,x'(0)=0$を入れる場合は、
ic2(% , t=0 , x(0)=0 , diff(x(t),t)=0);
2
g t
x(t) = - ----
2
$\displaystyle{x\left(t\right)=-{{g\,t^2}\over{2}}}$
(3)$\displaystyle{ \dfrac{dx}{dt}=\dfrac{5x}{3t} }$
ode2( diff(x(t), t, 1) = 5*x(t)/(3*t), x(t), t );
5 log(t)
--------
3
x(t) = %c %e
radcan(%);・・・logなどの関数を簡単にする。
5/3
x(t) = %c t
