## Related questions with answers

Find a general solution. Check your answer by substitution. y''+36y=0

Solutions

VerifiedThis is a problem in which we are given the ODE and asked to find the $\color{#4257b2}\text{general solution}$. To do so we will need to use the characteristic equation to find a root, afterwards we will use the correct form of a general solution to finish the problem. **Do you know what is the characteristic equation for this ODE?**

The characteristic equation is

$\lambda ^2 + 36 = 0$

Let's find the roots of the characteristic equation.

$\begin{align*} \lambda ^2 - (6i)^2 = 0 &\iff (\lambda - 6i) ( \lambda + 6i)= 0 \\ & \iff \lambda - 6i = 0 \quad \vee \quad \lambda + 6i = 0 \\ & \iff \lambda =6i \quad \vee \quad \lambda = -6i \end{align*}$

So, it has the complex conjugate roots:

$\lambda _1 = 6i \; (= a + ib) \quad \wedge \quad \lambda _2 = -6i \; (= a - ib)$

A basis is:

$\begin{align*} \begin{cases} & \phi _1 (x) = e^{ax} \cos (bx) = e^{0\cdot x} \cos 6x = \cos 6x \\ &\phi _2 (x) =e^{ax} \sin (bx) = e^{0\cdot x} \sin 6x = \sin 6x \end{cases} \end{align*}$

The general solution is:

$\begin{align*} y & = c_1 \phi _1 (x) + c_2 \phi _2 (x) \\ & = \boxed{{\color{#4257b2}{ c_1\cos 6x + c_2\sin 6x }}} \end{align*}$

${\color{#c34632} {y''+36y=0}}\;\;\;\; \Rightarrow \;\; \textbf{Homogeneous 2nd Order D.E , With Constant Coeff.}$

$\Rightarrow \textbf{Form :}\;\; \boxed{\;ay''+by'+cy=0\;}$

$Where\; : \;a\;,\;b\;,\;c \;\; \Rightarrow Constants$

$In\; order\; to\; solve\; this\; D.E \;, \;we\; need \;to\; get\; the\; "characterestic\; equation\;" by\; replacing$

$\boxed{y'' \;\; \Rightarrow \;\; m^{2}\;\;\;,\;\;\; y'\;\; \Rightarrow \;\; m\;\;\;,\;\;\;y\;\; \Rightarrow \;\; 1}$

$\textbf{Then Characterestic Eq :}\;\; \Rightarrow \;\;{\color{#c34632} {m^{2}+36=0}}$

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

$m_{1}=6\;\;\;,\;\;\;m_{2}=6$

$\therefore \;\; \textbf{Solution is : }\; {\color{#c34632} {y=C_{1}e^{m_{1}x}+C_{2}e^{m_{2}x}}}\;\;\; \Rightarrow \;\; \textbf{For }$

$\therefore \boxed{\;\color{#4257b2} {y=C_{1}e^{6x}+C_{2}e^{-6x}}\;}$

$r^2+36=0$

characteristic equation

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