- #1
Andrea94
- 21
- 8
- Homework Statement
- Solve a differential equation to obtain trajectory
- Relevant Equations
- Solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u##
I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##.
The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients:
which has general solution ##u_c=A cos\theta + B sin\theta##.
Now, I use variation of parameters to obtain the general solution to the nonhomogeneous equation. The form of the particular solution is ##u_p=u_1y_1+u_2y_2## where:
So that
The particular solution is then
Combining ##u_c## and ##u_p## to obtain the general solution yields
I ran the initial nonhomogeneous DE through an online solver, and got this exact same solution. However, in my book the answer is given as:
Where ##C## and ##\delta## are integration constants.
Any help on where I have gone wrong here? I hope I have provided all the required information in this post.
The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients:
##\frac{du^2}{d\theta ^2}+u=0##
which has general solution ##u_c=A cos\theta + B sin\theta##.
Now, I use variation of parameters to obtain the general solution to the nonhomogeneous equation. The form of the particular solution is ##u_p=u_1y_1+u_2y_2## where:
##u'_1=\frac{W_1}{W}##
##u'_2=\frac{W_2}{W}##
##W=y_1y'_2-y'_1y_2=cos^2\theta+sin^2\theta##
##W_1=-\frac{GM}{h^2} sin\theta##
##W_2=\frac{GM}{h^2} cos\theta##
##u'_2=\frac{W_2}{W}##
##W=y_1y'_2-y'_1y_2=cos^2\theta+sin^2\theta##
##W_1=-\frac{GM}{h^2} sin\theta##
##W_2=\frac{GM}{h^2} cos\theta##
So that
##u_1 = -\frac{GM}{h^2}\int sin\theta d\theta = \frac{GM}{h^2} cos\theta +C_1##
##u_1 = \frac{GM}{h^2}\int cos\theta d\theta = \frac{GM}{h^2} sin\theta +C_2##
##u_1 = \frac{GM}{h^2}\int cos\theta d\theta = \frac{GM}{h^2} sin\theta +C_2##
The particular solution is then
##u_p=(\frac{GM}{h^2} cos\theta +C_1)*cos\theta+(\frac{GM}{h^2} sin\theta +C_2)*sin\theta##
##u_p=\frac{GM}{h^2}+C_1cos\theta+C_2sin\theta##
##u_p=\frac{GM}{h^2}+C_1cos\theta+C_2sin\theta##
Combining ##u_c## and ##u_p## to obtain the general solution yields
##u=\frac{GM}{h^2}+C_1cos\theta+C_2sin\theta+A cos\theta+Bsin\theta##
##u=\frac{GM}{h^2}+c_1cos\theta+c_2sin\theta##
##u=\frac{GM}{h^2}+c_1cos\theta+c_2sin\theta##
I ran the initial nonhomogeneous DE through an online solver, and got this exact same solution. However, in my book the answer is given as:
##u = C cos(\theta+\delta)+\frac{GM}{h^2}##
Where ##C## and ##\delta## are integration constants.
Any help on where I have gone wrong here? I hope I have provided all the required information in this post.