How to solve a differential equation graphically?

Question

I want to solve the differential equation

$y'(t)=ty(t)+1$, $y(0)=1$

graphically in the interval $[0,1]$.

edit: Graphically means here, how to draw it with pen and paper.

But how do I do this, without any concrete information about $y(t)$. Am I supposed to get a vector field, or a 'standard function' (like the plot of $e^x$).

Can you explain how to solve such an equation graphically?

Thanks in advance.

Answer 1

As said in the comments, drawing a direction field is probably the way to go. For the direction field of \begin{equation} y'(t)=ty(t) \end{equation} (without taking the boundary condition into account), we can use a computer (if you want the R code, feel free to ask) to obtain

I assume you are able to find the general solution to this problem. Are you able to see how to use the boundary condition together with the direction field from above to come up with a specific solution to this problem?