Q12E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Derive the system (7) in the special case when $\mathit{n}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{3}$. [Hint: To determine the last equation, require that $\mathit{I}\mathbf{.}\left[y_p\right]\mathbf{=}\mathit{g}$  and use the fact that ${\mathit{y}}_{\mathbf{1}}\mathbf{,}{\mathit{y}}_{\mathbf{2}}$, and   satisfy the corresponding homogeneous equation.]

Step-by-Step Solution

Answer

The three functions  $V_1,V_2$ and $V_3$   that satisfy the systems are given as;

 $$\begin{gathered} V_1^{\text{'}}{y}_1+V_2^{\text{'}}{y}_2+V_3^{\text{'}}{y}_3=0 \\ {V}_1^{\text{'}}{y}_1^{\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}}=0 \\ {V}_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}=g. \end{gathered}$$

Step 1: Determine the three unknown function

Consider the differential equation

 $y\text{'}\text{'}\text{'}\left(x\right)+p_{1}y\text{'}\text{'}\left(x\right)+p_{2}y\text{'}\left(x\right)+p_{3}y\left(x\right)=g\left(x\right)$

where the coefficient functions  $p_1,p_2$ and $p_3$  as well as $g$  are continuous on$(a,b)$ . To find a particular solution to the given equation we need to know a fundamental solution set  $\left\{y_1,y_2,y_3\right\}$ for the corresponding homogeneous equation

 $y\text{'}\text{'}\text{'}\left(x\right)+p_{1}y\text{'}\text{'}\left(x\right)+p_{2}y\text{'}\left(x\right)+p_{3}y\left(x\right)=0.$

Therefore, a general solution to this homogeneous equation is

 $y_h\left(x\right)=c_1{y}_1\left(x\right)+c_2{y}_2\left(x\right)+c_3{y}_3\left(x\right)$

where   $c_1,c_2$and $c_3$  are arbitrary constants. In the method of variation of parameters, we assume there exists a particular solution to the given equation of the form

 $y_p\left(x\right)=V_1\left(x\right)y_1\left(x\right)+V_2\left(x\right)y_2\left(x\right)+V_3\left(x\right)y_3\left(x\right)$

and we try to determine the functions  $V_1\left(x\right),V_2\left(x\right)$ and $V_3\left(x\right)$. There are three unknown functions so we will need three equations to determine them. Differentiating  $y_p\left(x\right)$ gives us

 $y_p^{\text{'}}=\left(V_1{y}_1^{\text{'}}+V_2{y}_2^{\text{'}}+V_3{y}_3^{\text{'}}\right)+\left(V_1^{\text{'}}{y}_1+V_2^{\text{'}}{y}_2+V_3^{\text{'}}{y}_3\right)\text{. }$

To prevent second derivatives of the unknowns $V_1,V_2$  and  $V_3$ from entering the formula $y_p^{\text{'}\text{'}}$  we impose the condition

 $V_1^{\text{'}}{y}_1+V_2^{\text{'}}{y}_2+V_3^{\text{'}}{y}_3=0$

In the same manner, we impose the next condition

$V_1^{\text{'}}{y}_1+V_2^{\text{'}}{y}_2+V_3^{\text{'}}{y}_3=0$

Step 2: Determine the solution by using three equations.

Finally, the third condition that we impose is that   $y_p$satisfies the given equation.

 $$\begin{gathered} y_p^{\text{'}\text{'}\text{'}}\left(x\right)+p_1{y}_p^{\text{'}\text{'}}\left(x\right)+p_2{y}_p^{\text{'}}\left(x\right)+p_3{y}_p\left(x\right)=g\left(x\right) \\ \left(V_1{y}_1^{\text{'}\text{'}\text{'}}+V_2{y}_2^{\text{'}\text{'}\text{'}}+V_3{y}_3^{\text{'}\text{'}\text{'}}\right)+\left(V_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}\right)+p_1\left(V_1{y}_1^{\text{'}\text{'}}+V_2{y}_2^{\text{'}\text{'}}+V_3{y}_3^{\text{'}\text{'}}\right) \\ +p_2\left(V_1{y}_1^{\text{'}}+V_2{y}_2^{\text{'}}+V_3{y}_3^{\text{'}}\right)+p_3\left(V_1{y}_1+V_2{y}_2+V_3{y}_3\right)=g\left(x\right) \\ {V}_1\left(y_1^{\text{'}\text{'}\text{'}}+p_1{y}_1^{\text{'}\text{'}}+p_2{y}_1^{\text{'}}+p_3{y}_1\right)+V_2\left(y_2^{\text{'}\text{'}\text{'}}+p_1{y}_2^{\text{'}\text{'}}+p_2{y}_2^{\text{'}}+p_3{y}_2\right) \\ +V_3\left(y_3^{\text{'}\text{'}\text{'}}+p_1{y}_3^{\text{'}\text{'}}+p_2{y}_3^{\text{'}}+p_3{y}_3\right)+\left(V_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}\right)=g\left(x\right) \\ {V}_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}=g\left(x\right) \end{gathered}$$

So, using the previous conditions and the fact that   $y_1,y_2$and   $y_3$are solutions to the homogenous equation we get;

 $V_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}=g.$

Therefore, we seek three functions $V_1,V_2$  and $V_3$  that satisfy the system;

 $$\begin{gathered} V_1^{\text{'}}{y}_1+V_2^{\text{'}}{y}_2+V_3^{\text{'}}{y}_3=0 \\ {V}_1^{\text{'}}{y}_1^{\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}}=0 \\ {V}_1^{\text{'}}{y}_1^{\text{'}\text{'}}+V_2^{\text{'}}{y}_2^{\text{'}\text{'}}+V_3^{\text{'}}{y}_3^{\text{'}\text{'}}=g \end{gathered}$$