Second Order Nonhomogeneous Equation

T

tomrja

New member
Joined
Mar 25, 2011
Messages
2
Show that y=c[sub:31a4h8da]1[/sub:31a4h8da]cosx + c[sub:31a4h8da]2[/sub:31a4h8da]sinx + xsinx + (cosx)ln(cosx) is the general solution of y''+y=secx on (-pi/2,pi/2). Explain carefully.

I started off by setting y''+y=0 and verifying that cosx and sinx are solutions of the homogeneous function. I don't know where to go from here. Do I set secx equal to zero. Thank you for your help!
 
Have you studied Variation of Parameters yet?. It is fairly easy process for this one.

You have the roots of the auxiliary equation \(\displaystyle m^{2}+1=0\)

The roots are \(\displaystyle -i, \;\ i\). Meaning, the complementary function is \(\displaystyle y_{c}=C_{1}\underbrace{cos(x)}_{\text{y1}}+C_{2}\overbrace{sin(x)}^{\text{y2}}\)

\(\displaystyle f(x)=sec(x)\)

The general solution can be found by finding the Wronskian's (determinants):

\(\displaystyle W=\begin{vmatrix}y_{1}&y_{2}\\y_{1}^{'}&y_{2}^{'}\end{vmatrix}=\begin{vmatrix}cos(x)&sin(x)\\-sin(x)&cos(x)\end{vmatrix}=1\)

\(\displaystyle W_{1}=\begin{vmatrix}0&y_{2}\\ f(x)&y_{2}^{'}\end{vmatrix}=\begin{vmatrix}0&sin(x)\\ sec(x)&cos(x)\end{vmatrix}=-tan(x)\)

\(\displaystyle W_{2}=\begin{vmatrix}y_{1}&0\\ y_{1}^{'}&f(x)\end{vmatrix}=\begin{vmatrix}cos(x)&0\\ -sin(x)&sec(x)\end{vmatrix}=1\)

Now, compute \(\displaystyle u_{1}^{'}=\frac{W_{1}}{W}, \;\ u_{2}^{'}=\frac{W_{2}}{W}\)

Integrate to get \(\displaystyle u_{1}, \;\ u_{2}\). This will be the particular solution, \(\displaystyle y_{p}\)

Then, you should see the solution. Put it together to get the general solution, \(\displaystyle y=y_{c}+y_{p}\)
 
Thank you for your help! I don't think we went over how to solve these like the way you describe. We did the Wronskian, but I'm not sure how you got the W[sub:g37359mq]1[/sub:g37359mq] and W[sub:g37359mq]2[/sub:g37359mq] and everything after that. Is there a more simplified version of this?

Thanks again!
 
but I'm not sure how you got the W[sub:we3fmeb5]1[/sub:we3fmeb5] and W[sub:we3fmeb5]2[/sub:we3fmeb5] and everything after that
Click to expand...
.

Look at my previous post. It is outlined. You build the W, W1, W2 (determinants) from the given information. Derivatives, f(x), etc.
 
You must log in or register to reply here.
Top