Question

Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note that this is not a constant coefficient differential equation, but it is linear. The theory of linear differential equations states that the dimension of the space of all homogeneous solutions equals the order of the differential equation, so that a fundamental solution set for this equation should have two linearly fundamental solutions.

• Assume that y = x^r is a solution. Find the resulting
characteristic equation for r. • Write the general solution of the
differential equation.

• Find the solution for y(1) = 2 and y′(1) = 1.

Answer #1

We get

Thus, the characteristic equation is

Since , the solutions to this equation are . Thus, the general solution is

Note that . If and then

Solving the equations

we get and . Hence, the solution is

B. a non-homogeneous differential equation, a complementary
solution, and a particular solution are given. Find a solution
satisfying the given initial conditions.
y''-2y'-3y=6 y(0)=3 y'(0) = 11 yc=
C1e-x+C2e3x
yp = -2
C. a third-order homogeneous linear equation and three linearly
independent solutions are given. Find a particular solution
satisfying the given initial conditions
y'''+2y''-y'-2y=0, y(0) =1, y'(0) = 2, y''(0) = 0
y1=ex, y2=e-x,,
y3= e-2x

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

Differential Equations problem
If y1= e^-x is a solution of the differential equation
y'''-y''+2y=0 . What is the general solution of the differential
equation?

y'''+2y''-4y'+8y= -5cos(x)-10sin(x)+16e2x
A.Solve the underlining homogenous equation, solve the
characteristic equations, write fundamental solutions
B. Find the particular solution? (use undetermined coefficients
method to find particular solution)

Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity do not differ by an integer. Use the method of
Frobenius to obtain to linearly independent series solutions about
x = 0. Form the general solution on (0, ∞)
3xy”+(2 – x)y’ – y = 0

Find the fundamental solution to the following differential
equation.
y''+y'-2y=0, t0=0

Given the LDE : 2y''' - 5y'' + y' - 6x y = 1 + x
lnx (1) ,
identify the Homogeneous LDE associated with (1).
Answer choices:
A) Both equations are correct .
B) None of these
C) 2y''' - 5y'' + y' - 6x y = 0
D) 2y''' - 5y'' + y' - 6x y = 1 - x lnx

The indicated functions are known linearly independent solutions
of the associated homogeneous differential equation on (0, ∞). Find
the general solution of the given nonhomogeneous equation.
x2y'' + xy' + y = sec(ln(x))
y1 = cos(ln(x)), y2 = sin(ln(x))

Use the differential equation
(1−x)y''−4xy'+5y=cosx(1-x)y′′-4xy′+5y=cosx
to answer the following questions:
What is the type of the differential equation?
Ordinary differential equation
Partial differential equation
Is the differential equation linear or nonlinear?
Linear
Nonlinear
What is the order of the differential equation?

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