I am facing a problem in solving a system of three equations for three unknown parameters. In order to have a clear understanding of the issue, I have given a detailed description of the origin of the said equations. The equations arise from a differential equation and its boundary conditions from one of the research articles as:
$\omega^2 L C \frac{d^2v(x)}{dx^2} + v(x) - V_{cc}+ V_R \sin(x+\phi) = 0$
Whose General solution is given in the normalized form as
$\frac{v(x)}{V_{cc}}=C_1\cos qx+C_2\sin qx+1+ \frac{q^2}{1-q^2}.\frac{V_R}{V_{cc}}\sin (x+\phi)$
Where
$q=\frac{1}{\omega\sqrt{LC}}$
and
$\omega C\frac{d v(x)}{d x} =i_{c}(x)$
And the boundary conditions are:
$i_{c}(\pi)=\frac{\pi V_{cc}-2V_R \cos\phi}{\omega L} $
and
$v(\pi)=0$
So once I apply boundary conditions I have these two following equations
$0=C_1\cos q\pi+C_2\sin q\pi+1- \frac{q^2}{1-q^2}.\frac{V_R}{V_{cc}}\sin (\phi)\\ i_{c}(\pi)=\frac{\pi V_{cc}-2V_R \cos\phi}{\omega L} = \omega C \begin{bmatrix}-C_1q \sin(q\pi)+C_2q \cos(q\pi)+\frac{q^2}{1-q^2}V_R\cos(\phi) \end{bmatrix}_{x = \pi}$
After substituting the boundary conditions I am solving the above two equations for C1 and C2 constants using "Solve " in Wolfram Mathematica. and I obtain the following
$C_1= -\cos(q\pi)-q \pi \sin(q\pi)+(\frac{q}{1-q^2})\frac{V_R}{V_{cc}} \times\begin{bmatrix} q \cos(q\pi) \sin \phi +(1-2q^2) \sin(q\pi) \cos \phi \end{bmatrix}$
$C_2= -\sin(q\pi)+q \pi \cos(q\pi)+(\frac{q}{1-q^2})\frac{V_R}{V_{cc}} \times \begin{bmatrix} q \sin(q\pi) \sin \phi -(1-2q^2) \cos(q\pi) \cos \phi \end{bmatrix}$
The value of C1 and C2 is back substituted in v(x) of the general solution of the differential equation obtained above. Then I have set of three equations as follows
$V_R= -\frac{1}{\pi} \int_{0}^{2\pi} v(x)\sin(x+\phi)dx$
$\begin{bmatrix}v(x)\end{bmatrix}_{x=2\pi}=0$ $$\begin{bmatrix}\frac{d v(x)}{d x}\end{bmatrix}_{x=2\pi}=0$$
I am trying to solve the above three equations for three unknown parameters $$\phi,q, V_R$$ whereas Vcc is a constant in my equations. The three unknown parameters are reported by the authors of the research article as
$\phi = -41.614\\ q=1.607\\ V_R= 0.9253 \times V_{cc}$
I am solving the above set of three equations using Solve in Wolfram Mathematica but I get the error message as Solve::nsmet: This system cannot be solved with the methods available to Solve. >>. The complete notebook code that I use in Mathematica to solve it is given below
Clear[x, y, A, q, b, c, ysol, ysolsimp]
A = \[Omega]^2 *L*C
b = Vcc
c = VR
eqn = A* y''[x] + y[x] - b + c *Sin[x + phi] == 0
Solut = DSolve[eqn, y, x]
ysol = y /. First@DSolve[eqn, y, x]
ysolsimp = ysol[x] /. {C[1] -> c1, C[2] -> c2 , (x/(Sqrt[C]* Sqrt[L]* \[Omega])) -> q*x } //
FullSimplify
diff = D[ysolsimp, x]
ic = \[Omega]*C*diff
icp = \[Omega]*C*diff /. x -> Pi
icL = (Pi*Vcc - 2*VR*Cos[phi])/(\[Omega]*L)
icp == icL
eq1 = (icp == icL)
eq2 = (ysol[x] /. {C[1] -> c1, C[2] -> c2, x -> \[Pi]} /. (\[Pi]/( Sqrt[C]* Sqrt[L]* \[Omega])) -> q*\[Pi] // FullSimplify) == 0
soleq1 = Solve[ eq1 && eq2, {c1, c2}] /. { (\[Pi]/(Sqrt[C]* Sqrt[L]* \[Omega])) -> q*\[Pi] } // FullSimplify
c1 = c1 /. soleq1
c2 = c2 /. soleq1
inty = \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2 \[Pi]\)]\(ysolsimp*\ Sin[x + phi] \DifferentialD]x\)\) // FullSimplify
fourintegral = (-1/\[Pi])*\!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2 \[Pi]\)]\(ysolsimp*\ Sin[x + phi] \[DifferentialD]x\)\) // FullSimplify
eq4 = (fourintegral == VR)
eq5 = ysol[x] /. {C[1] -> c1, C[2] -> c2 , (x/(Sqrt[C]* Sqrt[L]* \[Omega])) -> q*x , x -> 2 \[Pi]} // FullSimplify
eq5b = (eq5 == 0)
eq6 = D[ysolsimp, x]
eq6b = eq6 /. x -> 2*\[Pi]
eq6c = eq6b == 0
Solve[{eq4 , eq5b , eq6c}, {phi, q, VR}]
Solve::nsmet: This system cannot be solved with the methods available to Solve. >>
The complete Wolfram Mathematica Notebook is also attached to this post. Kindly help me to debug the error. How can I get the same answer as that of the research article? Thanking you.
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