Alexis Alexander

2022-03-23

Find the general solution

$y{}^{\u2033}+5{y}^{\prime}-6y=0$

Cody Hart

Beginner2022-03-24Added 11 answers

Given:

$y{}^{\u2033}+5y-6y=0$

To find roots of this differential equation, Let m be the root of this equation that is,

${m}^{2}+5m-6=0$

Simplifying for m,

${m}^{2}+5m-6=0$

${m}^{2}-m+6m-6=0$

$m(m-1)+6(m-1)=0$

$(m+6)(m-1)=0$

$m=-6$ or $m=1$

So the roots are,

${m}_{1}=-6,\text{}{m}_{2}=1$

Since, the general solution of second order linear differential equation is,

$y={C}_{1}{e}^{{m}_{1}\cdot x}+{C}_{2}{e}^{{m}_{2}\cdot x}$

Where ${m}_{1}$ and $m}_{2$ are real and distinct roots of second order linear differential equation and $C}_{1$, $C}_{2$ are constant.

So now,

The general solution of given differential equation is,

$y={C}_{1}{e}^{-6x}+{C}_{2}{e}^{\left(1\right)x}$

$y={C}_{1}{e}^{-6x}+{C}_{2}{e}^{x}$

Answer: The final answer is given below,

$y={C}_{1}{e}^{-6x}+{C}_{2}{e}^{x}$

Jeffrey Jordon

Expert2022-03-31Added 2607 answers

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