Consider the differential equation $\dfrac { d y } { d x } = ( 3 - y ) \cos x$. Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. The function $f$ is defined for all real numbers. (a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point $( 0, 1 )$. (b) Write an equation for the line tangent to the solution curve in part (a) at the point $( 0, 1 )$. Use the equation to approximate $f ( 0.2 )$. (c) Find $y = f ( x )$, the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$.
Consider the differential equation $\dfrac { d y } { d x } = ( 3 - y ) \cos x$. Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. The function $f$ is defined for all real numbers.\\
(a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point $( 0, 1 )$.\\
(b) Write an equation for the line tangent to the solution curve in part (a) at the point $( 0, 1 )$. Use the equation to approximate $f ( 0.2 )$.\\
(c) Find $y = f ( x )$, the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$.