# Questions in Section Differential Equations

Please help me solve this DE problem: (4x^2-y^2)dx-xydy=0

Use convolution theorem to find the Inverse Laplace transform of 1/((s+1)(s^2+1)).

A mass weighing 32 lb. stretches a spring 32 ft. There is a damping constant of 2 (lb*s)/(ft). There is a force of 4e^(t) lb. Find the general solution to the differential equation: m u''+gamma u'+ku=F(t); u(0)=0, u'(0)=0.

Use the Laplace Transform to solve the equation: y''-3y'+2y=u_5(t); y(0)=0, y'(0)=0.

Use the Laplace Transform to solve the initial-value problem: y''+2y'+y=delta(t-2); y(0)=0, y'(0)=0.

Find solution of the system of differential equations in matrix form using eigenvalues and eigenvectors: x'=([3,1],[1,3])x.

This question concerns the differential equation x(d^2y)/(dx^2)-(2x+1)(dy)/(dx)+(x+1)y=4xe^x, and the associated homogeneous differential equation x(d^2y)/(dx^2)-(2x+1)(dy)/(dx)+(x+1)y=0.

1. Show that y_1(x)=e^x is a solution of the homogeneous differential equation.
2. Use the method of reduction of order to show that a second linearly independent solution of the homogeneous differential equation is y_2(x)=x^2e^x.
3. Use the method of variation of parameters to find the general solution of the given nonhomogeneous differential equation [Hint: Write the differential equation in standard form!].

Show that y_1=1 and y_2=ln(x) are both solutions of the non-linear differential equation y''+(y')^2=0.

1. Is y_1+y_2 also a solution? Demonstrate.
2. Is c_1y_1+c_2y_2 also a solution? Demonstrate.

The functions f_1(x)=3x-1, f_2(x)=4x and f_3(x)=2x are linearly dependent. Show this by finding values of c_1, c_2 and c_3, not all zeros, such that c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0.

Use the Laplace transform to solve the differential equation: y''+y=\delta(t-\pi), y(0)=0, y'(0)=0.