Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$. It can be shown that $f ^ { \prime \prime } ( 1 ) = 4$. (a) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 2 )$. (b) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 2 )$. Show the work that leads to your answer. (c) Find the particular solution $y = f ( x )$ to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$.
Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$. It can be shown that $f ^ { \prime \prime } ( 1 ) = 4$.\\
(a) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 2 )$.\\
(b) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 2 )$. Show the work that leads to your answer.\\
(c) Find the particular solution $y = f ( x )$ to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$.