2階線型非斉次微分方程式

$ \ddot x+2h\omega\dot x+\omega^2x=g(t)

解

$ \ddot x+(\alpha+\beta)\dot x+\alpha\beta=g(t)

$ \implies x=Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinc}\left(\frac{\alpha-\beta}{2i}t\right)\quad\text{.for }\exist A,B\in\Complex

$ x=Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinhc}\left(\frac{\alpha-\beta}2t\right)\quad\text{.for }\exist A,B\in\Complex

$ x=Be^{-\alpha^* t}+At^{\llbracket\Im\alpha=0\rrbracket}e^{-\alpha t}+f(t)*te^{-\Re\alpha t}\operatorname{sinc}\left(\Im\alpha t\right)\quad\text{.for }\exist A,B\in\Complex

$ x=(At+B)e^{-\alpha t}+f(t)*te^{-\alpha t}\quad\text{.for }\exist A,B\in\Complex

解法

$ \underline{f(t)=Ae^{i\hat{\omega}t}+Be^{-i\hat{\omega}^*t}\quad\text{.for }\exist A,B\in\Complex\quad}

となる

$ \ddot x+2h\omega\dot x+\omega^2x=g(t)

$ \implies((i\xi)^2+2h\omega(i\xi)+\omega^2)\mathcal F(x)=\mathcal F(g)

$ \iff(i\xi-i\hat{\omega})(i\xi+i\hat{\omega}^*)\mathcal F(x)=\mathcal F(g)

$ \iff\mathcal F(x)=-\frac{1}{\xi-\hat{\omega}}\frac{1}{\xi+\hat{\omega}^*}\mathcal F(g)

$ \iff x(t)=-\mathcal F^{-1}\left(\frac{1}{\xi-\hat{\omega}}\frac{1}{\xi+\hat{\omega}^*}\mathcal F(g)\right)(t)

$ =-\mathcal F\left(\frac{1}{\xi-\hat{\omega}}\frac{1}{\xi+\hat{\omega}^*}\mathcal F^{-1}(\tau\mapsto g(-\tau))\right)(-t)

$ =-\frac1{\sqrt{2\pi}}\left(\mathcal F\left(\frac{1}{\xi-\hat{\omega}}\frac{1}{\xi+\hat{\omega}^*}\right)*(\tau\mapsto g(-\tau))\right)(-t)

$ =-\frac1{\sqrt{2\pi}}\left(\left(\tau\mapsto\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\frac{1}{-\xi+\hat{\omega}^*}\right)(-\tau)\right)*(\tau\mapsto g(-\tau))\right)(-t)

$ =-\frac1{\sqrt{2\pi}}\left(\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\frac{1}{-\xi+\hat{\omega}^*}\right)*g\right)(t)

$ =-\frac1{\sqrt{2\pi}}\left(e^{i\xi\hat{\omega}}\mathcal F\left(\frac{1}{-\xi}\frac{1}{-\xi+2\Re\hat{\omega}}\right)*g\right)(t)

$ =-\frac1{2\pi}\left(\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\right)*\mathcal F\left(\frac{1}{-\xi+\hat{\omega}^*}\right)*g\right)(t)

$ =-\frac1{2\pi}\left(ie^{i\xi\hat{\omega}}\sqrt{\frac\pi2}\mathrm{sgn}\xi*ie^{-i\xi\hat{\omega}^*}\sqrt{\frac\pi2}\mathrm{sgn}\xi*g\right)(t)

$ =\frac14\left(e^{i\xi\hat{\omega}}\mathrm{sgn}\xi*e^{-i\xi\hat{\omega}^*}\mathrm{sgn}\xi*g\right)(t)

これ以上は分解できない

$ \mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\frac{1}{-\xi+\hat{\omega}^*}\right)=\frac{1}{2\Re\hat{\omega}}\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}-\frac{1}{-\xi+\hat{\omega}^*}\right)

$ =\frac{1}{2\Re\hat{\omega}}\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\right)-\frac{1}{2\Re\hat{\omega}}\mathcal F\left(\frac{1}{-\xi+\hat{\omega}^*}\right)

$ =\frac{1}{2\Re\hat{\omega}}e^{i\xi\hat{\omega}}\mathcal F\left(\frac{1}{-\xi}\right)-\frac{1}{2\Re\hat{\omega}}e^{-i\xi\hat{\omega}^*}\mathcal F\left(\frac{1}{-\xi}\right)

$ =\frac{1}{2\Re\hat{\omega}}ie^{i\xi\hat{\omega}}\sqrt{\frac\pi2}\mathrm{sgn}\xi-\frac{1}{2\Re\hat{\omega}}ie^{-i\xi\hat{\omega}^*}\sqrt{\frac\pi2}\mathrm{sgn}\xi

$ =-\frac1{\Re\hat{\omega}}\sqrt{\frac{\pi}{2}}\mathrm{sgn}\xi e^{-\Im\hat{\omega}\xi}\sin(\Re\hat{\omega}\xi)

$ \therefore x_p(t)=-\frac1{\sqrt{2\pi}}\left(\mathcal F\left(\frac{1}{-\xi-\hat{\omega}}\frac{1}{-\xi+\hat{\omega}^*}\right)*g\right)(t)

$ =-\frac{i}{4\Re\hat{\omega}}(\mathrm{sgn}t)(e^{i\hat{\omega}t}-e^{-i\hat{\omega}^*t})*g(t)

$ =\frac1{2\Re\hat{\omega}}\mathrm{sgn}t\Im e^{i\hat{\omega}t}*g(t)

$ =\frac1{2\Re\hat{\omega}}\left(\int_0^\infty\Im e^{i\hat{\omega}(t-\tau)}g(\tau)\mathrm d\tau-\int_{-\infty}^0\Im e^{i\hat{\omega}(t-\tau)}g(\tau)\mathrm d\tau\right)

$ =\frac1{2\Re\hat{\omega}}\int_0^\infty\left(\Im e^{i\hat{\omega}(t-\tau)}g(\tau)-\Im e^{i\hat{\omega}(t+\tau)}g(-\tau)\right)\mathrm d\tau

$ =\frac1{2\Re\hat{\omega}}\Im \left(e^{i\hat{\omega}t}\int_0^\infty\left( e^{-i\hat{\omega}\tau}g(\tau)-e^{i\hat{\omega}\tau}g(-\tau)\right)\mathrm d\tau\right)

$ x_p(t)=\frac1{2\Re\hat{\omega}}\Im \left(e^{i\hat{\omega}t}\int_0^\infty\left( e^{-i\hat{\omega}\tau}g(\tau)\mathrm d\tau-e^{i\hat{\omega}\tau}g(-\tau)\right)\mathrm d\tau\right)

$ =\frac1{2\Re\hat{\omega}}\Im e^{i\hat{\omega}t}\int_0^\infty\left(e^{-i\hat{\omega}\tau}+e^{i\hat{\omega}\tau}\right)g(\tau)\mathrm d\tau

$ =\frac C{2\Re\hat{\omega}}\int_0^\infty\left(\Im e^{i\hat{\omega}(t-\tau)}+\Im e^{i\hat{\omega}(t+\tau)}\right)\Im e^{i\omega_s\tau}\mathrm d\tau

複素積分になって解けない！

まいった

実世界での積分が大変なのだから、Fourier変換後の世界で掛け算してから逆変換すればいい

$ \mathcal F(g)(\xi)=\frac{C}{2i}(\mathcal F(e^{i\omega_s t})(\xi)-\mathcal F(e^{-i\omega_s t})(\xi))

$ = \frac{C}{2i}(\mathcal F(1)(\xi-\omega_s)-\mathcal F(1)(\xi+\omega_s))

$ =\sqrt{2\pi}\frac{C}{2i}(\delta(\xi-\omega_s)-\delta(\xi+\omega_s))

$ \therefore x_p(t)=-\sqrt{2\pi}\mathcal F^{-1}\left(\frac{1}{\xi-\hat{\omega}}\frac{1}{\xi+\hat{\omega}^*}\mathcal F(g)\right)(t)

$ =-\sqrt{2\pi}\mathcal F^{-1}\left(\frac{C}{2i}\frac{\delta(\xi-\omega_s)-\delta(\xi+\omega_s)}{(\xi-\hat{\omega})(\xi+\hat{\omega}^*)}\right)(t)

$ =-\frac{C}{2i}\sqrt{2\pi}\mathcal F^{-1}\left(\frac{\delta(\xi-\omega_s)-\delta(\xi+\omega_s)}{(\xi-\hat{\omega})(\xi+\hat{\omega}^*)}\right)(t)

$ =-\frac{C}{2i}\sqrt{2\pi}\mathcal F^{-1}\left(\frac{\delta(\xi-\omega_s)-\delta(\xi+\omega_s)}{(\xi-\hat{\omega})(\xi+\hat{\omega}^*)}\right)(t)

$ =-\frac{C}{2i}\sqrt{2\pi}\mathcal F^{-1}\left(\frac{\delta(\xi-\omega_s)-\delta(\xi+\omega_s)}{(\xi-\hat{\omega})(\xi+\hat{\omega}^*)}\right)(t)

$ =-\frac{C}{2i}\frac{e^{i\omega_st}}{(\omega_s-\hat{\omega})(\omega_s+\hat{\omega}^*)}+\frac{C}{2i}\frac{e^{-i\omega_st}}{(-\omega_s-\hat{\omega})(-\omega_s+\hat{\omega}^*)}

$ =-\frac{C}{2i}\frac{e^{i\omega_st}}{(\omega_s-\hat{\omega})(\omega_s+\hat{\omega}^*)}+\frac{C}{2i}\frac{e^{-i\omega_st}}{(\omega_s+\hat{\omega})(\omega_s-\hat{\omega}^*)}

$ =-\frac{C}{2i}\frac{e^{i\omega_st}}{(\omega_s-\hat{\omega})(\omega_s+\hat{\omega}^*)}+\frac{C}{2i}\left(\frac{e^{i\omega_st}}{(\omega_s-\hat{\omega})(\omega_s+\hat{\omega}^*)}\right)^*

$ =-C\Im\frac{e^{i\omega_st}}{(\omega_s-\hat{\omega})(\omega_s+\hat{\omega}^*)}

$ x_p(t)=-\frac C{|Z|}\sin((\omega_s-\arg Z)t)

$ |Z|^2=(\omega_s^2-|\hat{\omega}|^2)^2+4(\Im\hat{\omega})^2\omega_s^2

$ \underline{\therefore x(t)=Ae^{i\hat{\omega}t}+Be^{-i\hat{\omega}^*t}-\frac C{|Z|}\sin((\omega_s-\arg Z)t)\quad\text{.for }\exist A,B\in\Complex\quad}

$ \ddot x+2h\omega\dot x+\omega^2x=g(t)

$ \iff(D-i\hat{\omega})(D+i\hat{\omega}^*)x=g(t)

$ \iff(D-i\hat{\omega})z=g(t)

$ \iff D(e^{-i\hat{\omega}t}z)=g(t)e^{-i\hat{\omega}t}

$ \implies D(e^{-i\hat{\omega}t}z)=\frac1{2i}\left(e^{i(\omega_s-\hat{\omega})t}-e^{i(-\omega_s-\hat{\omega})t}\right)

$ \iff e^{-i\hat{\omega}t}z(t)=\frac1{2i}\left(\frac1{i(\omega_s-\hat{\omega})}e^{i(\omega_s-\hat{\omega})t}-\frac1{i(-\omega_s-\hat{\omega})}e^{i(-\omega_s-\hat{\omega})t}\right)-\frac1{2i}\left(\frac1{i(\omega_s-\hat{\omega})}-\frac1{i(-\omega_s-\hat{\omega})}\right)+z(0)

$ = -\frac12\left(\frac1{\omega_s-\hat{\omega}}e^{i(\omega_s-\hat{\omega})t}+\frac1{\omega_s+\hat{\omega}}e^{i(-\omega_s-\hat{\omega})t}\right)-\frac12\left(\frac1{\omega_s-\hat{\omega}}+\frac1{\omega_s+\hat{\omega}}\right)+z(0)

$ \iff z(t)= -\frac12\left(\frac1{\omega_s-\hat{\omega}}e^{i\omega_st}+\frac1{\omega_s+\hat{\omega}}e^{-i\omega_st}\right)-\frac12\left(\frac1{\omega_s-\hat{\omega}}+\frac1{\omega_s+\hat{\omega}}\right)e^{i\hat{\omega}t}+z(0)e^{i\hat{\omega}t}

$ \iff D(e^{i\hat{\omega}^*t}x)= -\frac12\left(\frac1{\omega_s-\hat{\omega}}e^{i\omega_st}+\frac1{\omega_s+\hat{\omega}}e^{-i\omega_st}\right)e^{i\hat{\omega}^*t}-\frac12\left(\frac1{\omega_s-\hat{\omega}}+\frac1{\omega_s+\hat{\omega}}\right)e^{2i\Re\hat{\omega}t}+z(0)e^{2i\Re\hat{\omega}t}

$ \iff e^{i\hat{\omega}^*t}x(t)= -\frac12\left(\frac1{\omega_s-\hat{\omega}}e^{i(\omega_s+\hat{\omega}^*)t}+\frac1{\omega_s+\hat{\omega}}e^{i(-\omega_s+\hat{\omega}^*)t}\right)-\frac12\left(\frac1{\omega_s-\hat{\omega}}+\frac1{\omega_s+\hat{\omega}}\right)\frac{1}{2i\Re\hat{\omega}}(e^{2i\Re\hat{\omega}t}-1)+z(0)\frac{1}{2i\Re\hat{\omega}}(e^{2i\Re\hat{\omega}t}-1)

計算が不毛に思えてきた……

$ (D-i\hat{\omega})(D-(i\hat{\omega})^*)x_p=f(t)

$ \iff D(e^{-i\hat{\omega} t}(D-(i\hat{\omega})^*)x_p)=A_fe^{i(\omega_f-\hat\omega)t}+B_fe^{-i({\omega_f}^*+\hat\omega)t}

$ = D\frac{A_f}{i(\omega_f-\hat\omega)}e^{i(\omega_f-\hat\omega)t}\llbracket\omega_f\neq\hat\omega\rrbracket-D\frac{B_f}{i({\omega_f}^*+\hat\omega)}e^{-i({\omega_f}^*+\hat\omega)t}\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket+A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket

$ \implies(D+i\hat\omega^*)x_p=\frac{A_f}{i(\omega_f-\hat\omega)}e^{i\omega_ft}\llbracket\omega_f\neq\hat\omega\rrbracket-\frac{B_f}{i({\omega_f}^*+\hat\omega)}e^{-i{\omega_f}^*t}\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket+(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)te^{i\hat{\omega}t}

$ \iff D(e^{i\hat\omega^*t}x_p)=\frac{A_f}{i(\omega_f-\hat\omega)}e^{i(\omega_f+\hat\omega^*)t}\llbracket\omega_f\neq\hat\omega\rrbracket-\frac{B_f}{i({\omega_f}^*+\hat\omega)}e^{-i({\omega_f}^*-\hat\omega^*)t}\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket+(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)te^{i(\hat{\omega}+\hat\omega^*)t}

$ =-D\frac{A_f}{(\omega_f-\hat\omega)(\omega_f+{\hat\omega}^*)}e^{i(\omega_f+\hat\omega^*)t}\llbracket\omega_f\neq\hat\omega\rrbracket\llbracket\omega_f\neq-{\hat\omega}^*\rrbracket-D\frac{B_f}{({\omega_f}^*+\hat\omega)({\omega_f}^*-{\hat\omega}^*)}e^{-i({\omega_f}^*-\hat\omega^*)t}\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket{\omega_f}\neq\hat\omega\rrbracket

$ +\frac{A_f}{i(\omega_f-\hat\omega)}\llbracket\omega_f\neq\hat\omega=-{\omega_f}^*\rrbracket-\frac{B_f}{i({\omega_f}^*+\hat\omega)}\llbracket{\omega_f}^*\neq-\hat\omega=-{\omega_f}\rrbracket

$ +D(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)\frac{t-\frac1{2i\Re\hat\omega}}{2i\Re\hat\omega}e^{2i\Re\hat\omega t}\llbracket\Re\hat\omega\neq0\rrbracket

$ +D\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2\llbracket\Re\hat\omega=0\rrbracket

$ =-D\left(\frac{A_f}{(\omega_f-\hat\omega)(\omega_f+{\hat\omega}^*)}e^{i(\omega_f+\hat\omega^*)t}+\frac{B_f}{({\omega_f}^*+\hat\omega)({\omega_f}^*-{\hat\omega}^*)}e^{-i({\omega_f}^*-\hat\omega^*)t}\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ -\frac1{2i\Re\hat\omega}\left(A_f\llbracket\hat\omega=-\omega_f\rrbracket+B_f\llbracket\hat\omega=\omega_f\rrbracket\right)\llbracket\Re\omega_f\neq0\rrbracket

$ +D(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)\frac{t-\frac1{2i\Re\hat\omega}}{2i\Re\hat\omega}e^{2i\Re\hat\omega t}\llbracket\Re\omega_f\neq0\rrbracket

$ +D\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2\llbracket\Re\omega_f=0\rrbracket

$ \implies x_p=-\left(\frac{A_f}{(\omega_f-\hat\omega)(\omega_f+{\hat\omega}^*)}e^{i\omega_ft}+\frac{B_f}{({\omega_f}^*+\hat\omega)({\omega_f}^*-{\hat\omega}^*)}e^{-i{\omega_f}^*t}\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ -\frac1{2i\Re\hat\omega}\left(A_f\llbracket\hat\omega=-\omega_f\rrbracket+B_f\llbracket\hat\omega=\omega_f\rrbracket\right)te^{-i\hat\omega^*t}\llbracket\Re\omega_f\neq0\rrbracket

$ +(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)\frac{t-\frac1{2i\Re\hat\omega}}{2i\Re\hat\omega}e^{i\hat\omega t}\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2e^{-i\hat\omega^*t}\llbracket\Re\omega_f=0\rrbracket

$ +Ce^{-i\hat\omega^*t}\quad\text{.for }\exist C\in\Complex

$ \implies x_p=-\left(\frac{A_f}{(\omega_f-\hat\omega)(\omega_f+{\hat\omega}^*)}e^{i\omega_ft}+\frac{B_f}{({\omega_f}^*+\hat\omega)({\omega_f}^*-{\hat\omega}^*)}e^{-i{\omega_f}^*t}\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ +(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)\frac{t}{2i\Re\hat\omega}e^{i\hat\omega t}\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2e^{-i\hat\omega^*t}\llbracket\Re\omega_f=0\rrbracket

斉次解に含められる項を消した

$ A_f\frac{z}{w}+B_f\frac{z^*}{w^*}=\frac1{|w|^2}(A_fzw^*+B_fz^*w)

$ =\frac1{|w|^2}(f(t)\Re w-i(A_fz-B_fz^*)\Im w)

$ =\frac1{|w|^2}\left(f(t)\Re w-\frac{f'(t)+f(t)\Im\omega_f}{\Re\omega_f}\Im w\right)

$ =\frac1{|w|^2}\left(f(t)\left(\Re w-\frac{\Im\omega_f}{\Re\omega_f}\Im w\right)-f'(t)\frac{\Im w}{\Re\omega_f}\right)

$ =\frac1{|w|^2}\left(\frac{f(t)}{\Re\omega_f}\Re(w\omega_f)-f'(t)\frac{\Im w}{\Re\omega_f}\right)

$ =\frac1{|w|^2}\frac1{\Re\omega_f}\left(f(t)\Re(w\omega_f)-f'(t)\Im w\right)

$ =\frac1{|w|^2}\left(f(t)\left(\Re(\omega_f^2)+4\Im\omega_f\Im\hat\omega-|\hat\omega|^2\right)-2f'(t)\Im(\omega_f-\hat\omega)\right)

$ w=(\omega_f-\hat\omega)(\omega_f+{\hat\omega}^*)

$ ={\omega_f}^2-2i\omega_f\Im\hat\omega-|\hat\omega|^2

$ \Im w=2\Re\omega_f\Im\omega_f-2\Re\omega_f\Im\hat\omega=2\Re\omega_f\Im(\omega_f-\hat\omega)

$ \Re(w\omega_f)=\Re\omega_f\left(\Re(\omega_f^2)+4\Im\omega_f\Im\hat\omega-|\hat\omega|^2\right)

$ |w|^2=(|\omega_f|^2+|\hat\omega|)^2-4\Re(\omega_f\hat\omega)^2

$ f'(t)=i\omega_fA_fz-i{\omega_f}^*B_fz^*

$ = i\Re\omega_f(A_fz-B_fz^*)-\Im\omega_f(A_fz+B_fz^*)

$ =-f(t)\Im\omega_f+i\Re\omega_f(A_fz-B_fz^*)

$ \frac1w=\frac1{2\Re\hat\omega}\left(\frac1{w_1}-\frac1{w_2}\right)

$ A_f\frac{z}{w_1}+B_f\frac{z^*}{{w_1}^*} =\frac1{|w_1|^2}\frac1{\Re\omega_f}\left(f(t)\Re(w_1\omega_f)-f'(t)\Im w_1\right)

$ \iff x_p=-\frac1{|w|^2}\frac1{\Re\omega_f}\left(f(t)\Re(w\omega_f)-f'(t)\Im w\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ +\frac t{2i\Re\hat\omega}\left(A_fe^{i\hat\omega t}\llbracket\omega_f=\hat\omega\rrbracket+B_fe^{i\hat\omega t}\llbracket{\omega_f}^*=-\hat\omega\rrbracket-A_fe^{-i\hat\omega^*t}\llbracket\omega_f=-\hat\omega\rrbracket-B_fe^{-i\hat\omega^*t}\llbracket\hat\omega=\omega_f\rrbracket\right)\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2e^{-i\hat\omega^*t}\llbracket\Re\omega_f=0\rrbracket

$ \iff x_p=-\frac1{|w|^2}\left(f(t)\left(\Re(\omega_f^2)+4\Im\omega_f\Im\hat\omega-|\hat\omega|^2\right)-2f'(t)\Im(\omega_f-\hat\omega)\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ +\frac t{2i\Re\hat\omega}\left(A_fe^{i\hat\omega t}-B_fe^{-i\hat\omega^*t}\right)\llbracket\omega_f=\hat\omega\rrbracket\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac t{2i\Re\hat\omega}\left(B_fe^{i\hat\omega t}\llbracket{\omega_f}^*=-\hat\omega\rrbracket-A_fe^{-i\hat\omega^*t}\llbracket\omega_f=-\hat\omega\rrbracket\right)\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2e^{-i\hat\omega^*t}\llbracket\Re\omega_f=0\rrbracket

$ \iff x_p=-\frac1{|w|^2}\left(f(t)\left(\Re(\omega_f^2)+4\Im\omega_f\Im\hat\omega-|\hat\omega|^2\right)-2f'(t)\Im(\omega_f-\hat\omega)\right)\llbracket{\omega_f}^*\neq-\hat\omega\rrbracket\llbracket\omega_f\neq\hat\omega\rrbracket

$ -\frac t2\frac{f'(t)+f(t)\Im\omega_f}{(\Re\omega_f)^2}\llbracket\omega_f=\hat\omega\rrbracket\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac t{2i\Re\hat\omega}\left(B_fe^{i\hat\omega t}\llbracket{\omega_f}^*=-\hat\omega\rrbracket-A_fe^{-i\hat\omega^*t}\llbracket\omega_f=-\hat\omega\rrbracket\right)\llbracket\Re\omega_f\neq0\rrbracket

$ +\frac12(A_f\llbracket\omega_f=\hat\omega\rrbracket+B_f\llbracket{\omega_f}^*=-\hat\omega\rrbracket)t^2e^{-i\hat\omega^*t}\llbracket\Re\omega_f=0\rrbracket

複雑になってしまった

パス！

$ (D+\alpha)(D+\beta)x=f(t)

$ \iff D(e^{\alpha t}(D+\beta)x)=f(t)e^{\alpha t}

$ \iff D(e^{(\alpha-\beta)t}D(e^{\beta t}x))=f(t)e^{\alpha t}

$ \iff e^{(\alpha-\beta)t}D(e^{\beta t}x)=\left.D(e^{\beta t}x)\right|_{t=0}+\int_0^t f(u)e^{\alpha u}\mathrm du

$ \iff D(e^{\beta t}x)=\left.D(e^{\beta t}x)\right|_{t=0}e^{(\beta-\alpha)t}+e^{(\beta-\alpha)t}\int_0^t f(u)e^{\alpha u}\mathrm du

$ =\left.D(e^{\beta t}x)\right|_{t=0}e^{(\beta-\alpha)t}+e^{\beta t}\int_0^t f(u)e^{-\alpha(t-u)}\mathrm du

$ =\left.D(e^{\beta t}x)\right|_{t=0}e^{(\beta-\alpha)t}+e^{\beta t}f(t)*e^{-\alpha t}

$ \iff e^{\beta t}x=x(0)+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{e^{(\beta-\alpha)t}-1}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+t\llbracket\alpha=\beta\rrbracket\right)+\int_0^te^{\beta u}f(u)*e^{-\alpha u}\mathrm du

$ \iff x=x(0)e^{-\beta t}+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+te^{-\beta t}\llbracket\alpha=\beta\rrbracket\right)+e^{-\beta t}\int_0^te^{\beta u}f(u)*e^{-\alpha u}\mathrm du

$ =x(0)e^{-\beta t}+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+te^{-\beta t}\llbracket\alpha=\beta\rrbracket\right)+\int_0^t(f(u)*e^{-\alpha u})e^{-\beta(t-u)}\mathrm du

$ =x(0)e^{-\beta t}+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+te^{-\beta t}\llbracket\alpha=\beta\rrbracket\right)+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =\left(x(0)+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{-1}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+t\llbracket\alpha=\beta\rrbracket\right)\right)e^{-\beta t}+\left.D(e^{\beta t}x)\right|_{t=0}\frac{e^{-\alpha t}}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =\left(x(0)-\left.D(e^{\beta t}x)\right|_{t=0}\frac{1}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket\right)e^{-\beta t}+\left.D(e^{\beta t}x)\right|_{t=0}\left(\frac{1}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket+t\llbracket\alpha=\beta\rrbracket\right)e^{-\alpha t}+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =\left(x(0)-\frac{\beta x(0)+x'(0)}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket\right)e^{-\beta t}+(\beta x(0)+x'(0))\left(\frac{\llbracket\alpha\neq\beta\rrbracket}{\beta-\alpha}+t\llbracket\alpha=\beta\rrbracket\right)e^{-\alpha t}+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =\left(x(0)\llbracket\alpha=\beta\rrbracket-\frac{\alpha x(0)+x'(0)}{\beta-\alpha}\llbracket\alpha\neq\beta\rrbracket\right)e^{-\beta t}+(\beta x(0)+x'(0))\left(\frac{\llbracket\alpha\neq\beta\rrbracket}{\beta-\alpha}+t\llbracket\alpha=\beta\rrbracket\right)e^{-\alpha t}+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =(x(0)e^{-\beta t}+(\beta x(0)+x'(0))te^{-\alpha t})\llbracket\alpha=\beta\rrbracket+\left(-(\alpha x(0)+x'(0))e^{-\beta t}+(\beta x(0)+x'(0))e^{-\alpha t}\right)\frac{\llbracket\alpha\neq\beta\rrbracket}{\beta-\alpha} +f(t)*e^{-\alpha t}*e^{-\beta t}

$ \iff x=(Be^{-\beta t}+Ate^{-\alpha t})\llbracket\alpha=\beta\rrbracket+\left(Be^{-\beta t}+Ae^{-\alpha t}\right)\llbracket\alpha\neq\beta\rrbracket +f(t)*e^{-\alpha t}*e^{-\beta t}

$ \iff x=Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*e^{-\alpha t}*e^{-\beta t}

$ =Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinhc}\left(\frac{\alpha-\beta}2t\right)

$ \because e^{-\alpha t}*e^{-\beta t}=te^{-\frac{\alpha+\beta}{2}t}\left(\llbracket\alpha=\beta\rrbracket+\operatorname{sinhc}\left(\frac{\alpha-\beta}2t\right)\llbracket\alpha\neq\beta\rrbracket\right)

$ =te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinhc}\left(\frac{\alpha-\beta}2t\right)

$ \underline{\implies x=Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinhc}\left(\frac{\alpha-\beta}2t\right)\quad\text{.for }\exist A,B\in\Complex\quad}_\blacksquare

$ \underline{\iff x=Be^{-\beta t}+At^{\llbracket\alpha=\beta\rrbracket}e^{-\alpha t}+f(t)*te^{-\frac{\alpha+\beta}{2}t}\operatorname{sinc}\left(\frac{\alpha-\beta}{2i}t\right)\quad\text{.for }\exist A,B\in\Complex\quad}_\blacksquare

驚き

$ x=Be^{-i{\hat\omega}^*t}+At^{\llbracket\Re\hat\omega=0\rrbracket}e^{i\hat\omega t}+f(t)*te^{-\Im\hat\omega t}\operatorname{sinc}\left(-\Re\hat\omega t\right)\quad\text{.for }\exist A,B\in\Complex