미분방정식 전체 문제

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

${y}^{''}+6{y}^{'}+9y={e}^{-x}$

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

${y}^{''}+2{y}^{'}-8y=6{e}^{2x}$

Solve the differential equation using the method of variation of parameters.

$4{y}^{''}+y=2\sin x$

Solve the differential equation using the method of variation of parameters.

${y}^{''}-9y=8x$

Solve the differential equation using the method of variation of parameters.

${y}^{''}+y=\sec x,\quad 0<x<\pi /2$

Solve the differential equation using the method of variation of parameters.

${y}^{''}+4y=3\csc 2x,\quad 0<x<\pi /2$

Find the unique solution satisfying the differential equation and the initial conditions given, where ${y}_{p}(x)$ is the particular solution.

${y}^{''}-2{y}^{'}+y=12{e}^{x},$ ${\text{y}}_{p}(x)=6{x}^{2}{e}^{x},$ $y(0)=6,\,\,y'(0)=0$

Find the unique solution satisfying the differential equation and the initial conditions given, where ${y}_{p}(x)$ is the particular solution.

${y}^{''}-7{y}^{'}=4x{e}^{7x},$ ${y}_{p}(x)=\frac{2}{7}{x}^{2}{e}^{7x}-\frac{4}{49}x{e}^{7x},$ $y(0)=-1,\,\,y'(0)=0$

Find the unique solution satisfying the differential equation and the initial conditions given, where ${y}_{p}(x)$ is the particular solution.

${y}^{''}+y=\cos x-4\sin x,$ ${y}_{p}(x)=2x\cos x+\frac{1}{2}x\sin x,$ $y(0)=8,\,\,y'(0)=-4$

Find the unique solution satisfying the differential equation and the initial conditions given, where ${y}_{p}(x)$ is the particular solution.

${y}^{''}-5{y}^{'}={e}^{5x}+8{e}^{-5x},$ ${y}_{p}(x)=\frac{1}{5}x{e}^{5x}+\frac{4}{25}{e}^{-5x},$ $y(0)=-2,\,\,y'(0)=0$

In each of the following problems, two linearly independent solutions—${y}_{1}$ and ${y}_{2}$—are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume x > 0 in each exercise.

${x}^{2}{y}^{''}+2x{y}^{'}-2y=3x,$ ${y}_{1}(x)=x,\quad {y}_{2}(x)={x}^{-2}$

In each of the following problems, two linearly independent solutions—${y}_{1}$ and ${y}_{2}$—are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume x > 0 in each exercise.

${x}^{2}{y}^{''}-2y=10{x}^{2}-1,$ ${y}_{1}(x)={x}^{2},\quad {y}_{2}(x)={x}^{-1}$

A 100-g mass stretches a spring 0.1 m. Find the equation of motion of the mass if it is released from rest from a position 20 cm below the equilibrium position. What is the frequency of this motion?

A 400-g mass stretches a spring 5 cm. Find the equation of motion of the mass if it is released from rest from a position 15 cm below the equilibrium position. What is the frequency of this motion?

A mass that weighs 8 lb stretches a spring 6 inches. The system is acted on by an external force of $8\sin 8t$ lb. If the mass is pulled down 3 inches and then released, determine the position of the mass at any time.

A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by an external force of $8\sin (4t)$ lb. If the mass is pulled down 1 inch and then released, determine the position of the mass at any time.

Find the charge on the capacitor in an RLC series circuit where $L=40$ H, $R=30\Omega ,$ $C=1/200$ F, and $E(t)=200$ V. Assume the initial charge on the capacitor is 7 C and the initial current is 0 A.

Find the charge on the capacitor in an RLC series circuit where $L=2$ H, $R=24\Omega ,$ $C=0.005$ F, and $E(t)=12\sin 10t$ V. Assume the initial charge on the capacitor is 0.001 C and the initial current is 0 A.

A series circuit consists of a device where $L=1$ H, $R=20\Omega ,$ $C=0.002$ F, and $E(t)=12$ V. If the initial charge and current are both zero, find the charge and current at time t.

A series circuit consists of a device where $L=\frac{1}{2}$ H, $R=10\Omega ,$ $C=\frac{1}{50}$ F, and $E(t)=250$ V. If the initial charge on the capacitor is 0 C and the initial current is 18 A, find the charge and current at time t.