91Ó°ÊÓ

Find \(\operatorname{an} O D E y^{\prime \prime}+a y^{\prime}+b y=0\) for the given basis. $$e^{0.3 x}, e^{-3.5 x}$$

Find an ODE (1) for which the given functions are solutions. Show linear independence (a) by considering quotients, (b) by Theorem 2. $$e^{-k x} \cos \pi x, e^{-k x} \sin \pi x$$

Solve the given nonhomogeneous ODE by variation of parameters or undetermined coefficients. Give a general solution. (Show the details of your work.) \(\left(x^{8} D^{2}+x D+\left(x^{2}-\frac{1}{4}\right) l\right) y=x^{3 / 2} \cos x\) Hint. To find \(y_{1}, y_{2}\) set \(y=u x^{-1 / 2}\)

Find the motion of the mass-spring system modeled by the ODE and initial conditions. Sketch or graph the solution curve. In addition, sketch or graph the curve of \(y-y_{p}\) to see when the system practically reaches the steady state. $$\begin{aligned}&y^{\prime \prime}+6 y^{\prime}+8 y=4 \sin 2 t, \quad y(0)=0.7,\\\ &y^{\prime}(0)=-11.8\end{aligned}$$

Solve the initial value problem. State which rules you are using, Show each step of your calculation in detail? $$\begin{aligned} &y^{\prime \prime}+0.2 y^{\prime}+0.26 y=1.22 e^{0.5 x}\\\ &y(0)=3.5, \quad y^{\prime}(0)=0.35 \end{aligned}$$

Reduce to first order and solve (showing each step in detail). $$y^{\prime \prime}=k y^{\prime}$$

Reduce to first order and solve (showing each step in detail). $$y^{\prime \prime}=1+y^{\prime 2}$$

Find \(\operatorname{an} O D E y^{\prime \prime}+a y^{\prime}+b y=0\) for the given basis. $$1, e^{-3 x}$$

Solve the initial value problem. State which rules you are using, Show each step of your calculation in detail? $$\begin{aligned} &y^{\prime \prime}-y^{\prime}-12 y=144 x^{3}+12.5\\\ &y(0)=5, \quad y^{\prime}(0)=-0.5 \end{aligned}$$

Reduce to first order and solve (showing each step in detail). $$y y^{\prime \prime}=4 y^{\prime 2}$$