User:Egm6321.f11.team2.Xia/RP5.9

=R*5.9 -=

Given
Nonhomogeneous L2-ODE-CC

$$\displaystyle a_2 y'' + a_1 y' +a_0 y = f(t)$$ (5.9.1)

Find
Using special IFM to solve (5.9.1) for general $$\displaystyle f(t)$$

Solution
Suppose the integrating factor has the form $$\begin{align} h(t,y) \end{align}$$

Multiply the integrating factor to the 5.9.1.

$$\displaystyle h(t,y) \cdot \left[ {{a}_{2}}y''+{{a}_{1}}y'+{{a}_{0}}y \right] = h(t,y) \cdot \left[{{f}_{\left( t \right)}}\right] $$  (5.9.2)

rewrite the 5.9.2 to check the exactness condition.

$$\displaystyle F(x,y,y',y) = \underbrace{h(t,y){a}_{2}}_{f(x,y,p)}y+ \underbrace{h(t,y){{a}_{1}}p+ h(t,y) {{a}_{0}}y - h(t,y)\cdot f(t)}_{g(x,y,p)} $$  (5.9.3)

where $$\begin{align} p:= y' \end{align} $$

therefore,

$$\displaystyle f(x,y,p) = h(t,y){a}_{2} $$

$$\displaystyle g(x,y,p) = h(t,y){{a}_{1}}p+ h(t,y) {{a}_{0}}y - h(t,y)\cdot f(t)$$

There are two relations for the 2nd condition of exactness.

$$\displaystyle {f}_{tt}+2p{f}_{ty}+{p}^{2}{f}_{yy}={g}_{tp}+p{g}_{yp}-{g}_{y} $$  (5.9.4)

$$\displaystyle {f}_{tp}+p{f}_{yp}+2{f}_{y}={g}_{pp} $$  (5.9.5)

From (5.9.5), we have

$$\displaystyle {h}_{y}=0$$

 (5.9.6) and from 5.9.4

$$\displaystyle {h}_{tt}{a}_{2}-{h}_{t}{a}_{1}+h{a}_{0}=0 $$  (5.9.7) So

$$\displaystyle {a}_{2}h''-{a}_{1}h'+{a}_{0}h=0 $$  (5.9.8)

thus, assuming $$ \displaystyle h(t)={e}^{\alpha t} $$

From 5.9.8, we have

$$\displaystyle {a}_{2}{\alpha}^{2}-{a}_{1}\alpha +{a}_{0}=0 $$  (5.9.10) then $$\displaystyle \frac{{e}^{-\alpha t}}{2\alpha a_{2}-a_{1}}\left({e}^{(\frac{a_{1}}{a_{2}}-2\alpha)t}\,\int{e}^{-\alpha t}\, f\left(t\right)dt-\int{e}^{\alpha t}\, f\left(t\right)dt\right)$$  ( 5.9.11) If $$\displaystyle a_{1}^{2}-4a_{0}a_{2}\neq0$$, substitute the solution of (5.9.10) to (5.9.11), we have

$$\displaystyle y(t)={C}_{1}\,{e}^{-\frac{\left(\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}+{a}_{1}\right)\, t}{2\,{a}_{2}}}+{C}_{2}\,{e}^{\frac{\left(\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}-{a}_{1}\right)\, t}{2\,{a}_{2}}+}$$

$$\displaystyle \frac{{e}^{-\frac{\left(\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}+{a}_{1}\right)\, t}{2\,{a}_{2}}}}{\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}}\left({e}^{\frac{\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}\, t}{{a}_{2}}}\,\int{e}^{-\frac{\left(\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}-{a}_{1}\right)\, t}{2\,{a}_{2}}}\, f\left(t\right)dt-\int{e}^{\frac{\left(\sqrt{{a}_{1}^{2}-4\,{a}_{0}\,{a}_{2}}+{a}_{1}\right)\, t}{2\,{a}_{2}}}\, f\left(t\right)dt\right)$$