Solve the boundary value problem [latex]y^{\prime\prime}+2y=0,\quad y(0)=0,\quad y(\pi)=0.[/latex] (15)

The general solution of the differential equation is again given by equation (8), [latex]y=c_{1}\cos\left({\sqrt{2}}\,x\right)+c_{2}\sin\left({\sqrt{2}}\,x\right).\qquad\qquad\qquad\qquad\qquad\qquad(8)[/latex] [latex]y=c_{1}\cos\left({\sqrt{2}}x\right)+c_{2}\sin\left({\sqrt{2}}x\right).[/latex] The first boundary condition requires that [latex]c_{1}[/latex] = 0, and the second boundary condition leads to [latex]c_{2}\sin\left({\sqrt{2}}\pi\right)=0[/latex]. Since [latex]\sin\left({\sqrt{2}}\pi\right)\neq 0[/latex], it follows that [latex]c_{2}[/latex] = 0 also. Consequently, y = 0 for all x is the only solution of the problem (15). This example illustrates that a homogeneous boundary value problem may have only the trivial solution y = 0.

Question 10.1.3: Solve the boundary value problem y′′ + 2y = 0, y(0) = 0, y(π......

Elementary Differential Equations and Boundary Value Problems [1529447]

Solve the boundary value problem

$y^{\prime\prime}+2y=0,\quad y(0)=0,\quad y(\pi)=0.$ (15)

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The general solution of the differential equation is again given by equation (8),

$y=c_{1}\cos\left({\sqrt{2}}\,x\right)+c_{2}\sin\left({\sqrt{2}}\,x\right).\qquad\qquad\qquad\qquad\qquad\qquad(8)$

$y=c_{1}\cos\left({\sqrt{2}}x\right)+c_{2}\sin\left({\sqrt{2}}x\right).$

The first boundary condition requires that $c_{1}$ = 0, and the second boundary condition leads to $c_{2}\sin\left({\sqrt{2}}\pi\right)=0$ . Since $\sin\left({\sqrt{2}}\pi\right)\neq 0$ , it follows that $c_{2}$ = 0 also. Consequently, y = 0 for all x is the only solution of the problem (15). This example illustrates that a homogeneous boundary value problem may have only the trivial solution y = 0.