I am not able to solve the above problem. I don't think my equations are wrong but I am not able to solve them. Please help me with this.
The answer given in my workbook is 3.93cost(3t+59.9)
Method A
Method B
This is not a homework problem. I was practicising problems when I came across this one.
While I agree with Chu that you can save yourself some pain by using phasors/Fourier transform, it is possible to solve these equations.
$$\begin{align} &v_x)\quad &\frac{v_s-v_x}{1} + \frac{v_1-v_x}{2} + \frac{1}{3}\frac{d(v-v_x)}{dt} &= 0 \\ &v_1)\quad &\frac{v_x - v_1}{2} - i_1 &= 0 \\ &v)\quad &\frac{1}{3}\frac{d(v_x-v)}{dt} + i_2 - \frac{v}{5} &= 0\\ &tf)\quad &v &= 4v_1\\ & & i_1 &= 4i_2 \end{align}$$
You can eliminate \$\frac{1}{3}\frac{d(v-v_x)}{dt}\$ using the third equation by substituting it in the first equation.
$$\frac{1}{3}\frac{d(v-v_x)}{dt} = i_2 - \frac{v}{5}$$
You can then solve all equations but the third to all unknowns except \$v\$ to find that (I used a CAS like Maxima to solve it)
$$\begin{align} i_1 &= \frac{20v_s-9v}{55} \\ i_2 &= \frac{20v_s-9v}{220} \\ v_1 &= \frac{v}{4} \\ v_x &= \frac{160v_s - 17v}{220} \end{align}$$
which allows us to plug \$i_2\$ and \$v_x\$ back in:
$$\begin{align} \frac{1}{3}\frac{d(v-v_x)}{dt} &= i_2 - \frac{v}{5} \\ &\Downarrow \\ 79\frac{dv}{dt} + 53v &= 80\cos(3t) - 640\sin(3t) \end{align}$$
The homogeneous solution is not important, as we're only interested in the steady-state solution. The particular solution is given by
$$\begin{align} v(t) &= \frac{77960}{29489}\cos(3t) -\frac{7480}{29489}\sin(3t) \\ &= 2.656\cos(3t + 5.48^\circ) \end{align}$$
Now, I'm not sure how you would get the reference answer of your book. When I verified using LTSpice, it seemed to support this answer rather than the one from your book.
You appear to be doing a transient analysis; this problem calls for complex notation. Represent the capacitor's reactance as: \$\small -j\:X_C=-\large\frac{j}{\omega C}\small =-j\$, let the source voltage be: \$\small V_S=4+j0=4\$, and then use nodal analysis.
I calculated v using phasor analysis using y-parameter. Please help me in how to write time domain representation of v.
