(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$$\text{y''+0.6y'+0.09y=0, y(0)=2.2, y'(0)=0.14, }e^{-0.3x}\text{, }xe^{-0.3x}$$

(a) Verify that $y$ is a solution of the ODE.

(b) Determine from y the particular solution of the IVP.

(c) Graph the solution of the IVP.

$$y'+4y=1.4$$

$$y=ce^{-4x}+0.35$$

$$y(0)=2$$

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$$4x^2y''-3y=0, y(1)=-3, y(1)=0, x^{3/2},x^{-1/2}$$

Solve the initial value problem. Check that your answer satisfies the ODE as well as the initial conditions. (Show the details of your work.)

$y^{\prime \prime}-9 y=0, y(0)=-2, y^{\prime}(0)=-12$

Fill in the table for each equation.

$$y=50(2.2)^x$$

$$y=350(1.7)^x$$

Find an ODE for which the given function form a basis of solutions.

$e^x, e^{2 x}, e^{3 x}$

Find the curve through the origin in the xy-plane which satisfies y''=2y' and whose tangent at the origin has slope 1.

Show that the given functions are linearly independent. $\sin x, \cos x$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.

$$f _ { 1 } ( x ) = 0 , \quad f _ { 2 } ( x ) = x , \quad f _ { 3 } ( x ) = e ^ { x }$$

Reduce to first order and solve, showing each step in detail. x²y''-5xy'+9y=0, y_1=x³

Using the methods in reduction of order if a solution is known, reduce to first order and solve (showing each step of your calculation in detail):

$$x^2 y^{\prime \prime}-5 x y^{\prime}+9 y=0, \quad y_1=x^3$$

Reduce to first order and solve, showing each step in detail.

$$\text {x²y''-5xy'+9y=0, }y_1\text{=x³}$$

Lake Erie has a water volume of about 450 km³ and a flow rate (in and out) of about 175 km² per year. If at some instant the lake has pollution concentration p=0.04%, how long, approximately, will it take to decrease it to p/2, assuming that the inflow is much cleaner, say, it has pollution concentration p/4, and the mixture is uniform (an assumption that is only imperfectly true)? First guess.

Solve the ODE by integration or by remembering a differentiation formula

$$y'+2 \sin(2πx)=0$$