User:Egm4313.s12.team8.colocar/R1.6

Falling Stone (Sec. 1.1)
$$ \displaystyle y''=g=const $$

Parachutist (Sec. 1.2)
$$ \displaystyle mv'=mg-bv^2 $$

Out-flowing Water (Sec.1.3)
$$ \displaystyle h'=-k\sqrt{h} $$

Vibrating Mass on a Spring (Sec. 2.4,2.8)
$$ \displaystyle my''+ky=0 $$

Beats of a vibrating system (Sec. 2.8)
$$ \displaystyle y''+w_0^2y= \cos wt$$

Current I in RLC circuit (Sec. 2.9)
$$ \displaystyle LI''+RI'+ \frac 1 c I=E' $$

Deformation of a Beam (Sec. 3.3)
$$ \displaystyle EIy^{iv} = f(x) $$

Pendulum (Sec. 4.5)
$$ \displaystyle L \theta''+g \sin \theta = 0 $$

Falling Stone (Sec. 1.1)
Order: 2nd Linearity: Linear Superposition: Yes The Homogeneous Solution, $$ \displaystyle y_h''=0 $$ plus the Particular Solution, $$ \displaystyle y_p''=g $$ yields $$ \displaystyle y_h+y_p=g $$ which is the same as $$ \displaystyle (y_h+y_p)= \bar y=g $$. Therefore, the Superposition Principle can be applied.

Parachutist (Sec. 1.2)
Order: 1st Linearity: Nonlinear Superposition: No

Rewriting the original equation with $$ \displaystyle \mathit{v}$$ on one side yields $$ \displaystyle mv'+bv^2=mg $$

The Homogeneous Solution is $$ \displaystyle mv_h'+bv_h^2=0 $$.

The Particular Solution is $$ \displaystyle mv_p'+bv_p^2=mg $$. Summing both solutions yields $$ \displaystyle mv_h'+bv_h^2 + mv_p'+bv_p^2 = m(v_h+v_p)' + b(v_h^2+v_p^2) = mg $$.

$$ \bar v = v_h+v_p $$, however, $$ v_h^2+v_p^2 \neq (v_h+v_p)^2 $$

Thus, the Superposition Principle CANNOT be applied.

Out-flowing Water (Sec.1.3)
Order: 1st Linearity: Nonlinear Superposition: No

The Homogeneous Solution $$ \displaystyle h_h'=-k\sqrt{h_h} $$

plus the Particular Solution $$ \displaystyle h_p'=-k\sqrt{h_p} $$

yields $$ \displaystyle h_h'+h_p'=-k\sqrt{h_h}-k\sqrt{h_p}$$

or $$ \displaystyle (h_h+h_p)'= \bar h' = -k(h_h+h_p)^{1/2}$$.

Yet, $$ \displaystyle -k(h_h^{1/2}+h_p^{1/2}) \neq -k(h_h+h_p)^{1/2}$$.

Therefore, The Superposition Principle CANNOT be applied.

Vibrating Mass on a Spring (Sec. 2.4,2.8)
Order: 2nd Linearity: Linear Superposition: Yes

The Homogeneous Solution $$ \displaystyle my_h''+ky_h=0 $$

plus the Particular Solution $$ \displaystyle my_p''+ky_p=0 $$

yields $$ \displaystyle m(y_h+y_p)''+k(y_h+y_p)=0 $$.

or $$ \displaystyle m \bar y''+k \bar y=0 $$.

Therefore, the Superposition Principle applies.

Beats of a vibrating system (Sec. 2.8)
Order: 2nd Linearity: Linear Superposition: Yes

The Homogeneous Solution is $$ \displaystyle y_h''+w_0^2y_h=0 $$

The Particular Solution is $$ \displaystyle y_p''+w_0^2y_p= \cos wt $$ Summing both solutions yields $$ \displaystyle (y_h+y_p)'' + w_0^2(y_h+y_p) = \cos wt $$

We know $$ \displaystyle \bar y = y_h+y_p $$,

so $$ \displaystyle \bar y'' + w_0^2 \bar y = \cos wt $$

Therefore, the Superposition Principle can be applied.

Current I in RLC circuit (Sec. 2.9)
Order: 2nd Linearity: Linear Superposition: Yes

The Homogeneous Solution is $$ \displaystyle LI_h''+RI_h'+ \frac 1 c I_h=0 $$

The Particular Solution is $$ \displaystyle LI_p''+RI_p'+ \frac 1 c I_p=E' $$ Summing both solutions yields $$ \displaystyle L(I_h+I_p)''+R(I_h+I_p)'+ \frac 1 c (I_h+I_p)=E' $$

We know $$ \displaystyle \bar I = I_h+I_p $$,

so $$ \displaystyle L \bar I''+R \bar I'+ \frac 1 c \bar I=E' $$

Therefore, the Superposition Principle can be applied.

Deformation of a Beam (Sec. 3.3)
Order: 4th Linearity: Linear Superposition: Yes

The Homogeneous Solution $$ \displaystyle EIy_h^{iv} = 0 $$

plus the Particular Solution $$ \displaystyle EIy_p^{iv} = f(x) $$

yields $$ \displaystyle EI(y_h^{iv}+y_p^{iv}) = EI(y_h+y_p)^{iv} = f(x) $$.

Knowing $$\bar y = y_h+y_p$$, we have

$$ \displaystyle EI(\bar y)^{iv} = f(x) $$.

Thus, the Superposition Principle can be applied.

Pendulum (Sec. 4.5)
Order: 2nd Linearity: Nonlinear Superposition: No

The Homogeneous Solution $$ \displaystyle L \theta_h''+g \sin \theta_h = 0 $$

plus the Particular Solution $$ \displaystyle L \theta_p''+g \sin \theta_p = 0 $$

yields $$ \displaystyle L ( \theta_h+ \theta_p)+g{( \sin \theta_h+ \sin \theta_p)} = L ( \theta_h+ \theta_p)''+g{( \sin \theta_h+ \sin \theta_p)} = 0 $$

But, $$ g{( \sin \theta_h+ \sin \theta_p)} \neq g \sin ( \theta_h + \theta_p) $$

Therefore, the Superposition Principle CANNOT be applied.