- #1

karush

Gold Member

MHB

- 3,269

- 5

[a] Find the general solution of $y^\prime + 3y=t+e^{-2x}\quad \dfrac{dy}{dx}+f(x)y=g(x)$

\$\begin{array}{lll}

\textsf{Similarly} & \dfrac{dy}{dx}+Py=Q\\

\textsf{hence} & \mu(x)=\exp\left(\int f(x)\,dx\right)\\

\textsf{then} & \mu^\prime(x)=\exp\left(\int f(x)\,dx\right)f(x) \\

\textsf{then} & \mu(x)+y'=\mu(x)g(x)\\

\textsf{integrating factor} & \mu(x)=e^{3x}\\

\textsf{multiplying} & e^{3x}y'+3e^{3x}y=xe^{3x}+e^{x}\\

\textsf{rewriting the LHS} & \dfrac{d}{dx}\left(e^{3x}y\right)=xe^{3x}+e^{x}\\

\end{array}$

\$\begin{array}{lll}

\textsf{Similarly} & \dfrac{dy}{dx}+Py=Q\\

\textsf{hence} & \mu(x)=\exp\left(\int f(x)\,dx\right)\\

\textsf{then} & \mu^\prime(x)=\exp\left(\int f(x)\,dx\right)f(x) \\

\textsf{then} & \mu(x)+y'=\mu(x)g(x)\\

\textsf{integrating factor} & \mu(x)=e^{3x}\\

\textsf{multiplying} & e^{3x}y'+3e^{3x}y=xe^{3x}+e^{x}\\

\textsf{rewriting the LHS} & \dfrac{d}{dx}\left(e^{3x}y\right)=xe^{3x}+e^{x}\\

\end{array}$

**determine how the solution behave as $t \to \infty$**

$ce^{-3x}+\dfrac{x}{3}-\dfrac{1}{9}+e^{-2x}$

y is asymptotic to $\dfrac{t}{3} −\dfrac{1}{9} \textit{ as } t \to \infty$

ok i think this is correct just could be worded better

maybe some typos

suggestions, complaints, or ?

$ce^{-3x}+\dfrac{x}{3}-\dfrac{1}{9}+e^{-2x}$

y is asymptotic to $\dfrac{t}{3} −\dfrac{1}{9} \textit{ as } t \to \infty$

ok i think this is correct just could be worded better

maybe some typos

suggestions, complaints, or ?

Last edited: