Q2E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$$\begin{gathered} \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\sqrt{\mathbf{x}}\mathbf{y}\mathbf{=}\mathbf{sinx} \\ \mathbf{y}\left(\pi \right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\left(\pi \right)\mathbf{=}\mathbf{11}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\pi \right)\mathbf{=}\mathbf{3} \end{gathered}$$

Step-by-Step Solution

Answer

Hence, the largest interval  $\left[\mathbf{0}\mathbf{,} \infty \right).$

Step 1: Solve the given equation

The given equation is $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\sqrt{\mathbf{x}}\mathbf{y}\mathbf{=}\mathbf{sinx}.$

 

Compare with the standard form of a linear differential equation,

 

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{p}\left(\mathbf{x}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{q}\left(\mathbf{x}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{r}\left(\mathbf{x}\right)\mathbf{y}\mathbf{=}\mathbf{s}\left(\mathbf{x}\right)$

 

Therefore,

 

$\mathbf{r}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\sqrt{\mathbf{x}}\mathbf{,} \mathbf{s}\left(\mathbf{x}\right)\mathbf{=}\mathbf{sinx}$

Step 2: Check the continuity

$\mathbf{r}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\sqrt{\mathbf{x}}$ is continuous for all $\mathbf{x}⩾\mathbf{0}$.

 

$\mathbf{s}\left(\mathbf{x}\right)\mathbf{=}\mathbf{sinx}$ is continuous in $R$.

Step 3:The largest interval (a, b)

Now overall r and s is continuous in $\forall \mathbf{x}\in \left[\mathbf{0}\mathbf{,} \infty \right).$

 

The initial condition is defined at ${\mathbf{x}}_{\mathbf{0}}\mathbf{=}\mathbf{\pi }.$

 

And $\mathbf{\pi }\in \left[\mathbf{0}\mathbf{,} \infty \right).$

 

Hence, the largest interval  $\left[\mathbf{0}\mathbf{,} \infty \right).$