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## 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.