Chapter 31

Find the general solution to each differential equation. $$y^{\prime \prime}-6 y^{\prime}+5 y=0$$

The displacement \(x\) of an object at \(t\) seconds is given by \(x=3.75 \cos 182 t\) in. Find (a) the period and (b) the amplitude of this motion.

Solve each second-order differential equation. With Algebraic Expressions $$y^{\prime \prime}-4 y=12$$

Solve each equation for \(y .\) Try some by calculator. $$y^{\prime \prime}=5$$

Find the general solution to each differential equation. $$2 y^{*}-5 y^{\prime}-3 y=0$$

Solve each second-order differential equation. With Algebraic Expressions $$y^{\prime \prime}+y^{\prime}-2 y=3-6 x$$

Find the general solution to each differential equation. $$y^{*}-3 y^{\prime}+2 y=0$$

The equation of motion of a certain wood block bobbing in water is \(x^{\prime \prime}+225 x=0,\) where \(x\) is in centimeters and \(t\) is in seconds. The initial conditions are \(x=0\) and \(v=15.0 \mathrm{cm} / \mathrm{s}\) at \(t=0 .\) Write an equation for \(x\) as a function of time.

Solve each second-order differential equation. With Algebraic Expressions $$y^{\prime \prime}-y^{\prime}-2 y=4 x$$

Solve each equation for \(y .\) Try some by calculator. $$y^{\prime \prime}=3 e^{x}$$

Find the general solution to each differential equation. $$y^{\prime \prime}+4 y^{\prime}-5 y=0$$

Solve each second-order differential equation. With Algebraic Expressions $$y^{\prime \prime}+y^{\prime}=x+2$$

Find the general solution to each differential equation. $$y^{*}-y^{\prime}-6 y=0$$

Solve each second-order differential equation. With Algebraic Expressions $$y^{\prime \prime}-4 y=x^{3}+x$$

Find the general solution to each differential equation. $$y^{\prime \prime}+5 y^{\prime}+6 y=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}+2 y^{\prime}-3 y=42 e^{4 x}$$

Find the general solution to each differential equation. $$5 y^{\prime \prime}-2 y^{\prime}=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}-y^{\prime}-2 y=6 e^{x}$$

Find the general solution to each differential equation. $$y^{\prime \prime}+4 y^{\prime}+3 y=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}+y^{\prime}=6 e^{x}+3$$

The equation of motion of a certain car shock absorber is given by \(x=2.50 e^{-3 t} \cos 55 t\) in., where \(t\) is in seconds. Make a graph of \(x\) versus \(t.\)

Find the general solution to each differential equation. $$6 y^{*}+5 y^{\prime}-6 y=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}-4 y=4 x-3 e^{x}$$

Find the general solution to each differential equation. $$y^{\prime \prime}-4 y^{\prime}+y=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}-y=e^{x}+2 e^{2 x}$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}-4 y^{\prime}+4 y=0$$

If \(R=10.5 \Omega, L=0.175 \mathrm{H}, C=1.50 \times 10^{-3} \mathrm{F},\) and \(e=175 \mathrm{sin} 55 t,\) find the amplitude of the steady-state current.

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}-6 y^{\prime}+9 y=0$$

Solve each second-order differential equation. With Exponential Expressions. $$y^{\prime \prime}-y=x e^{x}$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}-2 y^{\prime}+y=0$$

An object has a differential equation of motion given by \(x^{\prime \prime}+5 x^{\prime}+4 x=0\) with the initial conditions \(x_{0}=1.50 \mathrm{cm}\) and \(v_{0}=15.3 \mathrm{cm} / \mathrm{s} .\) Write an equation for \(x\) as a function of time.

Solve each second-order differential equation. With Trigonometric Expressions. $$y^{\prime \prime}+4 y=\sin 2 x$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$9 y^{*}-6 y^{\prime}+y=0$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}+4 y^{\prime}+4 y=0$$

Solve each second-order differential equation. With Trigonometric Expressions. $$y^{\prime \prime}+2 y^{\prime}+y=\cos x$$

If an RLC circuit has the values \(R=10 \Omega, L=0.200 \mathrm{H}, C=500 \mu \mathrm{F},\) and \(e=100 \sin \omega t,\) find the resonant frequency.

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$4 y^{2}+4 y^{\prime}+y=0$$

Solve each second-order differential equation. With Trigonometric Expressions. $$y^{\prime \prime}+4 y^{\prime}+4 y=\cos x$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}+2 y^{\prime}+y=0$$

Solve each second-order differential equation. With Trigonometric Expressions. $$y^{\prime \prime}+y=2 \cos x-3 \cos 2 x$$

Second-Order DE, Roots of Auxiliary Equation Real and Equal $$y^{\prime \prime}-10 y^{\prime}+25 y=0$$

Solve each second-order differential equation. With Trigonometric Expressions. $$y^{\prime \prime}+y=\sin x+1$$

Second-Order DE, Roots of Auxiliary Equation Not Real $$y^{\prime \prime}+4 y^{\prime}+13 y=0$$

Solve each second-order differential equation. With Exponential and Trigonometric Expressions. $$y^{\prime \prime}+y=e^{x} \sin x$$

Second-Order DE, Roots of Auxiliary Equation Not Real $$y^{\prime \prime}-2 y^{\prime}+2 y=0$$

Solve each second-order differential equation. With Exponential and Trigonometric Expressions. $$y^{\prime \prime}+y=10 e^{x} \cos x$$

Second-Order DE, Roots of Auxiliary Equation Not Real $$y^{\prime \prime}-6 y^{\prime}+25 y=0$$

Solve each second-order differential equation. With Exponential and Trigonometric Expressions. $$y^{\prime \prime}-4 y^{\prime}+5 y=e^{2 x} \sin x$$

Second-Order DE, Roots of Auxiliary Equation Not Real $$y^{\prime \prime}+2 y^{\prime}+2 y=0$$

Solve each second-order differential equation. With Exponential and Trigonometric Expressions. $$y^{\prime \prime}+2 y^{\prime}+5 y=3 e^{-x} \sin x-10$$