Circuit Theory/2Source Excitement/Node and Mesh

The first step is to convert everything to phasors and impedances in symbolic form if possible:
 * $$\mathbb{V}_1 = 5*(\sqrt{3} + j)$$
 * $$\mathbb{I}_1 = \frac{1-j}{2*\sqrt{2}}$$
 * $$Z_{C1} = \frac{1}{j \omega C_1} = -10j$$
 * $$Z_{C2} = \frac{1}{j \omega C_2} = -5j$$
 * $$Z_L = j \omega L_1 = L_2 = 1j$$

Node Analysis

 * $$\frac{\mathbb{V}_1 - \mathbb{V}_a}{R_1 + j \omega L_1} + \mathbb{I}_1 - \frac{\mathbb{V}_a - \mathbb{V}_b}{\frac{1}{j \omega C_1}} = 0 $$
 * $$\frac{\mathbb{V}_a-\mathbb{V}_b}{\frac{1}{j \omega C_1}} - \frac{\mathbb{V}_b}{R_3} - \frac{\mathbb{V}_b-\mathbb{V}_c}{\frac{1}{j \omega C_2}} = 0$$
 * $$\frac{\mathbb{V}_b-\mathbb{V}_c}{\frac{1}{j \omega C_2}} - \mathbb{I}_1 - \frac{\mathbb{V}_c}{j \omega L_2} = 0$$

Results using matlab:
 * $$\mathbb{V}_a = -6.5 - 7i \Rightarrow V_a(t) = 9.55 cos(1000t - 2.32)$$
 * $$\mathbb{V}_b = -3.08 - 3.31i \Rightarrow V_b(t) = 4.52 cos(1000t - 2.32)$$
 * $$\mathbb{V}_c = 0.327 + 0.385i \Rightarrow V_c(t) = 0.506 cos(1000t + 0.866)$$

Mesh Analysis

 * $$\mathbb{I}_1 R_2 + \frac{\mathbb{I}_1 + \mathbb{I}_2}{j \omega C_1} + \frac{\mathbb{I}_1 + \mathbb{I}_3}{j \omega C_2} - \mathbb{V}_{Is} = 0$$
 * $$\mathbb{V}_1 + \mathbb{I}_2 R_1 + \frac{\mathbb{I}_2 + \mathbb{I}_1}{j \omega C_1} + (\mathbb{I}_2 - \mathbb{I}_3) R_3 + \mathbb{I}_2 j \omega L_1 = 0$$
 * $$\frac{\mathbb{I}_3 + \mathbb{I}_1}{j \omega C_2} + \mathbb{I}_3 j \omega L_2 + (\mathbb{I}_3 - \mathbb{I}_2) R_3 = 0$$

Results using Matlab:
 * $$\mathbb{I}_2 = -0.239 + 0.234i \Rightarrow I_2(t)= 0.334 cos(1000t + 2.37 )$$
 * $$\mathbb{I}_3 = -0.238 + 0.235i \Rightarrow I_3(t) = 0.334 cos(1000t + 2.36)$$
 * $$\mathbb{V}_{Is} = 175 - 179i \Rightarrow V_{Is}(t) = 250 cos(1000t - 0.795)$$