Consider the differential equation $\frac { d y } { d x } = 3 x + 2 y + 1$. (a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. (b) Find the values of the constants $m , b$, and $r$ for which $y = m x + b + e ^ { r x }$ is a solution to the differential equation. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = - 2$. Use Euler's method, starting at $x = 0$ with a step size of $\frac { 1 } { 2 }$, to approximate $f ( 1 )$. Show the work that leads to your answer. (d) Let $y = g ( x )$ be another solution to the differential equation with the initial condition $g ( 0 ) = k$, where $k$ is a constant. Euler's method, starting at $x = 0$ with a step size of 1 , gives the approximation $g ( 1 ) \approx 0$. Find the value of $k$.
076$ because $f ( t ) < g ( t )$ for $0 \leq t < 1.617$ and $3 < t < 5.076$.
Consider the differential equation $\frac { d y } { d x } = 3 x + 2 y + 1$.
(a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$.
(b) Find the values of the constants $m , b$, and $r$ for which $y = m x + b + e ^ { r x }$ is a solution to the differential equation.
(c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = - 2$. Use Euler's method, starting at $x = 0$ with a step size of $\frac { 1 } { 2 }$, to approximate $f ( 1 )$. Show the work that leads to your answer.
(d) Let $y = g ( x )$ be another solution to the differential equation with the initial condition $g ( 0 ) = k$, where $k$ is a constant. Euler's method, starting at $x = 0$ with a step size of 1 , gives the approximation $g ( 1 ) \approx 0$. Find the value of $k$.