91Ó°ÊÓ

A 49-N weight is attached to the lower end of a coil spring suspended from the ceiling and having a spring constant of \(10 \mathrm{N} / \mathrm{m}\). The resistance in the spring-mass system is numerically equal to 10 times the instantaneous velocity. At \(t=0\) the weight is set in motion from a position \(0.6 \mathrm{m}\) below its equilibrium position by giving it a downward velocity of \(0.6 \mathrm{m} / \mathrm{s}\). At the end of \(\pi\) s, determine whether the mass is above or below the equilibrium position and by what distance.

Use power series to find the general solution of the differential equation. $$(1+x) y^{\prime \prime}-y=0$$

Solve the equations by the method of undetermined coefficients. $$y^{\prime \prime}-y^{\prime}-2 y=20 \cos x$$

Find the general solution to the given Euler equation. Assume \(x>0\) throughout. \(3 x^{2} y^{\prime \prime}+4 x y^{\prime}=0\)

Use power series to find the general solution of the differential equation. $$\left(1-x^{2}\right) y^{\prime \prime}-4 x y^{\prime}+6 y=0$$

Find the general solution of the given equation. $$9 y^{\prime \prime}-y=0$$

Solve the equations by the method of undetermined coefficients. $$y^{\prime \prime}+y=2 x+3 e^{x}$$

Find the general solution to the given Euler equation. Assume \(x>0\) throughout. \(x^{2} y^{\prime \prime}+6 x y^{\prime}+4 y=0\)

A 29.4-N weight stretches a spring 1.225 m. The spring-mass system resides in a medium offering a resistance to the motion equal to 18 times the instantaneous velocity. If the weight is released at a position \(0.6 \mathrm{m}\) above its equilibrium position with a downward velocity of \(0.9 \mathrm{m} / \mathrm{s},\) find its position relative to the equilibrium position 2 s later.

Find the general solution of the given equation. $$8 y^{\prime \prime}-10 y^{\prime}-3 y=0$$