User:Egm6321.f10.team3.franklin/Hwk5

(2.6)
I expanded my solution for 2.4,


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$$ \displaystyle \begin{align} y & =\frac\int_ – ^{t}{{{e}^{(\beta -\alpha )\tau }}(\int_ – ^{\tau }{{{e}^{\alpha s}}f(s)ds}})d\tau \\ & =\frac\int_ – ^{t}{{{e}^{(\beta -\alpha )\tau }}\left( \int{{{e}^{\alpha \tau }}f(\tau )d\tau }+{{k}_{1}} \right)d\tau } \\ & =\frac\left[ \int_ – ^{t}{{{e}^{(\beta -\alpha )\tau }}\left( \int{{{e}^{\alpha \tau }}f(\tau )d\tau } \right)d\tau }+{{k}_{1}}\int_ – ^{t}{{{e}^{(\beta -\alpha )\tau }}d\tau } \right] \\ & =\frac\left[ \int{{{e}^{(\beta -\alpha )t}}\left( \int{{{e}^{\alpha t}}f(t)dt} \right)dt}+{{k}_{2}}+\frac{\beta -\alpha }{{e}^{(\beta -\alpha )t}} \right] \\ & =\frac{(\beta -\alpha ){_{2}}}{{e}^{-\alpha t}}+\frac{{e}^{-\beta t}}+\frac{{{{{a}}}_{2}}}\int{{{e}^{(\beta -\alpha )t}}\left( \int{{{e}^{\alpha t}}f(t)dt} \right)dt} \end{align}$$ $$
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 * $$\displaystyle (Eqs. 5.6.1)
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Combine constants to form k1 and k2,


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$$ \displaystyle y={{C}_{1}}{{e}^{-\alpha t}}+{{C}_{2}}{{e}^{-\beta t}}+\frac\int{{{e}^{(\beta -\alpha )t}}\left( \int{{{e}^{\alpha t}}f(t)dt} \right)dt}$$ $$
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 * $$\displaystyle (Eq. 5.6.2)
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From above we seen the homog. and particular solutions,


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$$ \displaystyle \begin{align} & y^{1}={{e}^{-\alpha t}} \\ & y^{2}={{e}^{-\beta t}} \\ & {{y}_{P}}=\frac\int{{{e}^{(\beta -\alpha )t}}\left( \int{{{e}^{\alpha t}}f(t)dt} \right)dt} \\ \end{align}$$ $$
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 * $$\displaystyle (Eqs. 5.6.3)
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