Find a real general solution. Show the details of your work. x²y''+0.7xy'-0.1y=0

Find the general solution of the differential equation

$$\begin{gather*} x^{2}y'' - 3xy' + 4y = \ln x, \end{gather*}$$

by using the change of variable ${t = \ln x}$ to transform the equation into one with constant coefficients.

y''+ay'+by=0 for the given basis. cos 2πx, sin 2πx.

y''+ay'+by=0 for the given basis.

$$e^{2.6x}\text{, }e^{-4,3x}$$

Find the equilibrium solutions of the differential equation specified.

$$\frac{dy}{dt} =\frac{(t^{2}-1)(y^{2}-2)}{y^{2}-4}$$

Find a real general solution. Show the details of your work. (x²D²-xD+5l)y=0

Find the equilibrium solutions of the differential equation specified.

$$\frac{dy}{dt} =\frac{y+3}{1-y}$$

Find a general solution. Check your answer by substitution. y''+y'+3.25y=0

Solve (x ²D² -3xD +4I) y= 0

A proton in a cyclotron gains $\Delta K=2 e \Delta V$ of kinetic energy per revolution, where $\Delta V$ is the potential between the dees. Although the energy gai n comes in small pulses, the proton makes so many revolutions that it is reasonable to model the energy as increasing at the constant rate $P=d K / d t=\Delta K / T$, where T is the period of the cyclotron motion. This is power input because it is a rate of increase of energy. a. Find a n expression for r(t), the radius of a proton’s orbit in a cyclotron, in terms of m, e, B, P, and t. Assume that r = 0 at t = 0. Hint: Start by finding an expression for the proton’s kinetic energy in terms of r. b. A relatively small cyclotron is 2.0 m in diameter, uses a 0.55 T magnetic field, and has a 400 V potential difference between the dees. What is the power input to a proton, in W? c. How long does it take a proton to spiral from the center out to the edge?

Find a real general solution. Show the details of your work. 4x²y''+5y=0

Find a real general solution. Show the details of your work. xy''+2y'=0

Find the general solution of the non-homogeneous Cauchy–Euler equation

$$\begin{gather*} x^{2} y'' - 3xy' + 4y = x^2 \ln x, \,\,\,\,\, x > 0. \end{gather*}$$

(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.

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