4739.sagews

We want to solve the initial value problem

$$y'+y=e^{\cos t}$$

$$y(t_0)=y_0$$

Using the method of integrating factors we can find $\mu(t)=e^t$

$$e^ty'+e^ty=e^{t+\cos t}$$

which gives

$$[e^ty]'=e^{t+\cos t}$$

and

$$e^ty=\int e^{t+\cos t}\ dt + C$$

Finally

$$y=e^{-t}\int e^{t+\cos t}\ dt + Ce^{-t}$$

So all we need is the integral $\int e^{t+\cos t}\ dt$ and we are done

var('t')
integral(e^(t+cos(t)),t)

\newcommand{\Bold}[1]{\mathbf{#1}}\int e^{\left(t + \cos\left(t\right)\right)}\,{d t}

But the above tells us sage doesn't know how to find the integral abstractly. Which means it's a fairly sure bet we can't find it either. So instead of worrying we will use a numerical integral to find (and graph) solutions.

$$y(t)=e^{-t}\int_{t_0}^t e^{x+\cos x}\ dx + Ce^{-t}$$

Note that

$$y_0=y(t_0)=e^{-t_0}\int_{t_0}^{t_0} e^{x+\cos x}\ dx + Ce^{-t_0}=Ce^{-t_0}$$

Solving for $C$ gives us $C=e^{t_0}y_0$

Let's suppose that $t_0=0$ and plot some integral curves for $y$

var('t,u,x,y')
C=0
def f(t):
    return N(exp(-t)*(numerical_integral(exp(u+cos(u)),0,t)[0]+C))
p1=plot(f,(t,-10,10),ymin=-10,ymax=10)

C=7
p2=plot(f,(t,-10,10),ymin=-10,ymax=10,color='red')

C=-5
p3=plot(f,(t,-10,10),ymin=-10,ymax=10,color='green')

p=plot_slope_field(exp(cos(t))-y,(t,-10,10),(y,-10,10))
show(p+p1+p2+p3)