 # The general solution of the differential equationy^{n}-9y=0can b Harlen Pritchard 2021-09-16 Answered

The general solution of the differential equation
${y}^{n}-9y=0$
can be written in the form
$y\left(x\right)=A{e}^{{m}_{1}x}+B{e}^{{m}_{2}x}$,
where $A,B,{m}_{1},{m}_{2}\in R$ and ${m}_{1}>{m}_{2}$.
a) Solve the auxiliary equation, and enter the values of ${m}_{1}$ and ${m}_{2}$ in the boxes.
Do not enter decimals in your answers. Enter whole numbers, fractions or square roots as appropriate.
If you need to enter a square root, e.g. $\sqrt{x}$, enter this as $\sqrt{x}$.
${m}_{1}=$ ?
${m}_{2}=$ ?

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Step 1: To Find
We have to find the general solution of the differential equation $y-9y=0$ and write it in the form of $y\left(x\right)=A{e}^{{m}_{1}x}+B{e}^{{m}_{2}x}$.
Step 2: Calculation
$y9y=0$
Auxilary Equation: ${m}^{2}-9=0$
${m}^{2}-{3}^{2}=0$
$\left(m+3\right)\left(m-3\right)=0$
$m=-3,3$
Since roots of the auxiliary equation are distinct and $-3<3$, then the solution is
$y\left(x\right)=A{e}^{3x}+B{e}^{-3x}$
Hence ${m}_{1}=3$ and ${m}_{2}=-3$.