The General Solution of the DE y''+y'=tan(t) ?

[Ahmad Obeid]

Hello There, I hope I posted this in the right thread.
I've been struggling with solving this particular Differential Equation and just couldn't find any way to solve it completely..

1. Homework Statement
I am only required to find the general solution of the differential equation
y'' + y' = tan(t)

Homework Equations

Wronskian of two functions.
Characteristic Polynomial of a (homogeneous) Differential Equation.

The Attempt at a Solution

First I found the solution to the associated DE y''+y'=0 which gave me y1=1 and y2=e^-t
Thus the general solution is y= u1*y1 + u2*y2 where u1 and u2 are two functions of t to be determined.
Using the method of variation of parameters I ended up with u1 = -ln|cos(t)| + c1 (Note that the Wronskian of y1 & y2 is -e^-t )
However I ended up with u2 = -∫e^t*tant dt which is obviously unsolvable..
You can find attached my work and attempts.
Is there anything wrong? or is there another way around? like just writing the integral as an infinite series?
Thank you for your help!

 

[Ray Vickson]

Hello There, I hope I posted this in the right thread.
I've been struggling with solving this particular Differential Equation and just couldn't find any way to solve it completely..

1. Homework Statement
I am only required to find the general solution of the differential equation
y'' + y' = tan(t)

Homework Equations

Wronskian of two functions.
Characteristic Polynomial of a (homogeneous) Differential Equation.

The Attempt at a Solution

First I found the solution to the associated DE y''+y'=0 which gave me y1=1 and y2=e^-t
Thus the general solution is y= u1*y1 + u2*y2 where u1 and u2 are two functions of t to be determined.
Using the method of variation of parameters I ended up with u1 = -ln|cos(t)| + c1 (Note that the Wronskian of y1 & y2 is -e^-t )
However I ended up with u2 = -∫e^t*tant dt which is obviously unsolvable..
You can find attached my work and attempts.
Is there anything wrong? or is there another way around? like just writing the integral as an infinite series?
Thank you for your help!

Lots of problems have solutions that cannot be written in "closed form"; perhaps this is one of them. What I mean is that you can invent a new function $\Lambda(t) = \int_0^t \tan(s) e^s \, ds$ and can then express your answer in terms of $\Lambda(\cdot)$.

 

[Ahmad Obeid]

Oh so I just leave it as is? Thought there was some extra step I should make...
If only that y' was a y life would've been much easier haha
Thank you for your help !