Find the general solution of the given differential equation. x2y' + x(x + 2)y = ex y(x) = give the largest interval over which the general solution is defined. (think about the implications of any singular points. enter your answer using interval notation.) determine whether there are any transient terms in the general solution. (enter the transient terms as a comma-separated list; if there are none, enter none.)

Multiply both sides of the ODE by :

and notice that . So we can collapse the left side and write it as the derivative of a product:

We can't have , so (or , depending on the initial value) would be the largest interval over which the solution is valid.

As , we see the term rapidly approaching 0, so this is the only transient term.

step-by-step explanation:

let

we know that

the y-intercept is the y-coordinate of the point when the value of x is equal to zero

so

the y-coordinate of the point a is the y-intercept

the answer part a) is

the value of y-intercept is

part b) find the slope m

we know that

the value of the slope is equal to

substitute the values

therefore

the answer part b) is

the value of the slope is equal to

the general answer is

the slope is

the value of y-intercept is

c.

step-by-step explanation: