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Question Number 205172 by shunmisaki007 last updated on 12/Mar/24

$$x\left(t\right)=c+{tx}^{\prime }\left(t\right)\mid c\mathrm{is}\mathrm{constant}.$$$$\left(1\right)\mathrm{Find}x\left(t\right).$$$$\left(2\right)\mathrm{Find}x\left(t\right)\mathrm{when}x\left(0\right)=x_0.$$

Commented by lepuissantcedricjunior last updated on 12/Mar/24

$$\mathit{x}\left(\mathit{t}\right)=\mathit{c}+{\mathit{t}\mathit{x}}^{\prime }\left(\mathit{t}\right)\mathit{c}\in \mathbb{R}$$$$\mathit{o}\mathit{n}\mathit{a}$$$${\mathit{t}\mathit{x}}^{\prime }\left(\mathit{t}\right)-\mathit{x}\left(\mathit{t}\right)=-\mathit{c}$$$${\mathit{x}}_{\mathit{k}}={\mathit{e}}^{\int \frac{1}{\mathit{t}}\mathit{d}\mathit{t}}={\mathit{k}\mathit{e}}^{\mathit{l}\mathit{n}\mathit{t}}=\mathit{k}\mathit{t}$$$$\mathit{f}\mathit{a}\mathit{i}\mathit{s}\mathit{o}\mathit{n}\mathit{s}\mathit{v}\mathit{a}\mathit{r}\mathit{i}\mathit{e}\mathit{r}\mathit{l}\mathit{a}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathit{k}\left(\mathit{t}\right)$$$$\mathit{x}\left(\mathit{t}\right)=\mathit{k}\left(\mathit{t}\right)\mathit{t}$$$${\mathit{x}}^{\prime }\left(\mathit{t}\right)={\mathit{k}}^{\prime }\left(\mathit{t}\right)\mathit{t}+\mathit{k}\left(\mathit{t}\right)$$$${\mathit{t}\mathit{x}}^{\prime }\left(\mathit{t}\right)-\mathit{x}\left(\mathit{t}\right)=-\mathit{c}$$$${\mathit{t}}^2{\mathit{k}}^{\prime }\left(\mathit{t}\right)+\mathit{k}\left(\mathit{t}\right)\mathit{t}-\mathit{k}\left(\mathit{t}\right)\mathit{t}=-\mathit{c}$$$${\mathit{k}}^{\prime }\left(\mathit{t}\right)=-\frac{\mathit{c}}{{\mathit{t}}^2}=>\mathit{k}\left(\mathit{t}\right)=\frac{\mathit{c}}{\mathit{t}}+\mathit{m}\mathit{o}\mathit{u}\mathit{m}\in \mathbb{R}$$$${\mathit{x}}_{\mathit{m}}\left(\mathit{t}\right)=\mathit{m}\mathit{t}+\mathit{c}$$$$\mathit{x}\left(\mathit{o}\right)={\mathit{x}}_0$$$$=>{\mathit{x}}_{\mathit{m}}\left(0\right)=\mathit{m}\left(0\right)+\mathit{c}={\mathit{x}}_0$$$$=>\mathit{c}={\mathit{x}}_0$$$${\mathit{x}}_{\mathit{m}}\left(\mathit{t}\right)=\mathit{m}\mathit{t}+{\mathit{x}}_0\mathit{o}\mathit{u}\mathit{m}\in \mathbb{R}$$$${\mathit{x}}_{\mathit{m}}\left(\mathit{t}\right)=\mathit{m}\mathit{t}+{\mathit{x}}_0\mathit{o}\mathit{u}\mathit{m}\in \mathbb{R}$$$$.............lepuissant\mathit{D}r......................$$$$$$

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