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Question

# Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work. 9y''-30y'+25y=0, y(0)=3.3, y'(0)=10.0

Solution

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We have

$9y'' - 30y' + 25y = 0$

The characteristic equation,

$\begin{gather*} 9\lambda^{2}- 30\lambda + 25 = 0 \\ \text{ its roots:} \quad \lambda_{(1,2)}= \dfrac{5}{3} \end{gather*}$

where we have a case of a real double root, which means that the general solution of this Second-Order Linear ODE would be of the form

$\mathbf{y=\left( C_{1} + C_{2}x\right)e^{\lambda x}}$

Therefore,

\begin{align*} y&= \left( C_{1} + C_{2}x \right)e^{\frac{5}{3} x}\\ &= C_{1} e^{\frac{5}{3} x} + C_{2}xe^{\frac{5}{3} x} \end{align*}

and

$y'=\dfrac{5}{3}C_{1} e^{\frac{5}{3} x} + C_{2}e^{\frac{5}{3} x} + \dfrac{5}{3} C_{2} x e^{\frac{5}{3} x}$

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