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Solve the initial-value problem 2y" + 5y' - 3y = 0, y(0) = 3, y'(0) = 5.
Find the general solution to \(6 y^{\prime \prime}+18 y^{\prime}-24 y=0\). Enter your answer as \(y=\ldots\) In your answer, use \(c_{1}\) and \(c_{2}\) to denote arbitrary constants and \(x\) the independent variable. Enter \(c_{1}\) as \(\mathrm{c} 1\) and \(c_{2}\) as \(\mathrm{c} 2\).
a. Find a particular solution to the nonhomogeneous differential equation \(y^{\prime \prime}+4 y^{\prime}+5 y=-15 x+3 e^{-x}\).
The IVP$$ \begin{array}{c} \sin (t) \frac{d^{2} x}{d t^{2}}+\cos (t) \frac{d x}{d t}+\sin (t) x=\tan (t) \\ x(0.75)=12 \\ \left.\frac{d x}{d t}\right|_{0,75}=2 \end{array} $$has a unique solution defined on the interval