# Questions in Section Differential Equations

Find the inverse Laplace Transforms of `Y(s)=(s-2)/(s(s-4))`

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`.

- Show that `y_1(x)=e^x` is a solution of the homogeneous differential equation.
- 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`.
- 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`.

- Is `y_1+y_2` also a solution? Demonstrate.
- 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`.