Question

Verify that the indicated function is a solution of the given
dierentail

equation. In some cases assume an appropriate interval of validity
for the

solution.

y''+y'-12y=0; y=c1e^3x+c2e^-4x

Answer #1

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Suppose x=c1e−t+c2e^5t. Verify that x=c1e^−t+c2e^5t is a
solution to x′′−4x′−5x=0 by substituting it into the differential
equation. (Enter the terms in the order given. Enter c1 as c1 and
c2 as c2.)

Verify that the function ϕ(t)=c1e^−t+c2e^−2t is a solution of
the linear equation
y′′+3y′+2y=0
for any choice of the constants c1c1 and c2c2. Determine c1c1
and c2c2 so that each of the following initial conditions is
satisfied:
(a) y(0)=−1,y′(0)=4
(b) y(0)=2,y′(0)=0

Verify that the indicated function is a solution of the given
differentialequation.x2y′′−xy′+ 2y= 0;y=xcos (lnx), x >0

Verify that the given function is the solution of the initial
value problem.
1. A) x^3y'''-3x^2y''+6xy'-6y= -(24/x) y(-1)=0 y'(-1)=0
y''(-1)=0
y=-6x-8x^2-3x^3+(1/x)
C) xy'''-y''-xy'+y^2= x^2 y(1)=2 y'(1)=5 y''(1)=-1
y=-x^2-2+2e^(x-1-e^-(x-1))+4x

Find a second solution of the given differential equation. Use
reductionof order or Formula (4). Assume an appropriate interval of
validity.
(1 +x)y′′+xy′−y= 0 ; y1=x

Verify that the given functions form a fundamental set of
solutions of the differential equation on the indicated interval.
Form the general solution.
1.) y'' − 4y = 0; cosh 2x, sinh 2x, (−∞,∞)
2.) y^(4) + y'' = 0; 1, x, cos x, sin x (−∞,∞)

The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
1
(x)
dx (5) as instructed, to find a
second solution y2(x).
y'' + 36y = 0; y1 =
cos(6x)
y2 =
2) The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
1...

The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order, to find a
second solution dx **Please do not solve this via the
formula--please use the REDUCTION METHOD ONLY.
y2(x)= ??
Given: y'' + 2y' + y = 0; y1 =
xe−x

Verify that the function
y=x^2+c/x^2
is a solution of the differential equation
xy′+2y=4x^2, (x>0).
b) Find the value of c for which the solution satisfies the initial
condition y(4)=3.
c=

find a homogeneous linear differential equation with constant
coefficients whose general solution is given:
c1e^(x)sin4x+c2e^(x)cos4x

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