Find a general solution. Show the details of your calculation. (100D²-160D+64l)y=0

Find a general solution. Check your answer by substitution ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9. 4y''-25y=0

Find a general solution of the ODE y''+ω²y=r(t) with r(t) as given. Show the details of your work. r(t)=π/4|cost| if -π<t<π and r(t+2π)=r(t),|ω|≠0, 2, 4,...

For the differential equation

$$\begin{gather*} 6y'' - y' - y = 0, \end{gather*}$$

(a) Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two parameter family of trajectories

$$\begin{gather*} x = x_{1}i + x_{2}j = yi + y'j \end{gather*}$$

of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.

Reduce to first order and solve, showing each step in detail. yy''=3y'²

Find the general solution of the Cauchy–Euler equation

$$\begin{gather*} x^{2}y'' - 5xy' + 9y = 0 \end{gather*}$$

in ${x > 0}$.

Reduce to first order and solve, showing each step in detail. x²y''-5xy'+9y=0, y_1=x³

Show that F(x, y', y'')=0 can be reduced to first order in z=y' (from which y follows by integration). Give two examples of your own.

Find a general solution. Show the steps of derivation. Check your answer by substitution. y'sin2πx=πycos2πx

Solve the IVP. Show the steps of derivation, beginning with the general solution. y'cosh²x=sin²y, y(0)=½π

Solve the IVP. Show the steps of derivation, beginning with the general solution. y'=1+4y², y(1)=0

Solve the IVP. Show the steps of derivation, beginning with the general solution. xy'+y=0, y(4)=6

Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.

$$x^2D^2-4xD+ul)y=21x^{-4}$$

Find a general solution. Show the steps of derivation. Check your answer by substitution. xy'=y²+y (Set y/x=u)