91Ó°ÊÓ

The solution of the differential equationis

As per the superposition principle, when two or more waves overlap in space, the resultant disturbance is equal to the algebraic sum of the individual disturbances.

The solution of the differential equation $\frac{d^{2}Y}{d{t}^2}+2b\frac{dY}{dt}+{\mathrm{Ó\neg }}^{2}Y=F_1{e}^{i{w}_{1}t}+F_2{e}^{i{w}_{2}t}+F_3{e}^{i{w}_{3}t}$by using the principal of superposition can be written as:

$y_P=\left(\begin{array}{c}\frac{F_1}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }\text{'}{1}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{1}^2}}\sin \left(\mathrm{Ó\neg }\text{'}1t-\mathrm{φ}\right)+\frac{F_2}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }\text{'}{2}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{2}^2}}\sin \left({\mathrm{Ó\neg }}_2^{\text{'}}t-\mathrm{φ}\right)\\ +\frac{F_3}{\sqrt{{\left({\mathrm{Ó\neg }}^2-{{\mathrm{Ó\neg }}_3^{\text{'}}}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{3}^2}}\sin \left(\mathrm{Ó\neg }\text{'}3t-\mathrm{φ}\right)\end{array}\right)$

Now, if $\mathrm{Ó\neg }={\mathrm{Ó\neg }}_1^{\text{'}}$, then the above solution becomes:

$$\begin{gathered} y_P=\left(\begin{array}{c}\frac{F_1}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }\text{'}{1}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{1}^2}}\sin \left(\mathrm{Ó\neg }\text{'}1t-\mathrm{φ}\right)+\frac{F_2}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }\text{'}{2}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{2}^2}}\sin \left({\mathrm{Ó\neg }}_2^{\text{'}}t-\mathrm{φ}\right)\\ +\frac{F_3}{\sqrt{{\left({\mathrm{Ó\neg }}^2-{{\mathrm{Ó\neg }}_3^{\text{'}}}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{3}^2}}\sin \left(\mathrm{Ó\neg }\text{'}3t-\mathrm{φ}\right)\end{array}\right) \\ =\frac{F_1}{2b{\mathrm{Ó\neg }}_1}\sin \left({\mathrm{Ó\neg }}_1^{\text{'}}t-\mathrm{Ï•}\right)+\frac{F_2}{\sqrt{{\left({\mathrm{Ó\neg }}^2-{\mathrm{Ó\neg }}_2^2\right)}^2+4b^2\mathrm{Ó\neg }{2}^2}}\sin \left({\mathrm{Ó\neg }}_2^{\text{'}}t-\mathrm{Ï•}\right)+\frac{F_3}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }{3}^2\right)}^2+4b^2\mathrm{Ó\neg }{3}^2}}\sin \left({\mathrm{Ó\neg }}_3^{\text{'}}t-\mathrm{Ï•}\right) \end{gathered}$$

Thus, the solution of the differential equation $\frac{d^{2}Y}{d{t}^2}+2b\frac{dY}{dt}+{\mathrm{Ó\neg }}^{2}Y=F_1{e}^{i{\mathrm{Ó\neg }}_{2}t}+F_2{e}^{i{\mathrm{Ó\neg }}_{2}t}+F_3{e}^{i{\mathrm{Ó\neg }}_{3}t}$by using the principal of superposition is $y_p=\frac{F_1}{2b{\mathrm{Ó\neg }}_1}\sin \left({\mathrm{Ó\neg }}_1^{\text{'}}t-\mathrm{Ï•}\right)+\frac{F_2}{\sqrt{{\left({\mathrm{Ó\neg }}^2-{\mathrm{Ó\neg }}_2^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{2}^2}}\sin \left({\mathrm{Ó\neg }}_{2}t-\mathrm{Ï•}\right)+\frac{F_3}{\sqrt{{\left({\mathrm{Ó\neg }}^2-\mathrm{Ó\neg }{3}^2\right)}^2+4b^2\mathrm{Ó\neg }\text{'}{3}^2}}\sin \left({\mathrm{Ó\neg }}_{3}t-\mathrm{Ï•}\right)$