Determine which of the following pairs of functions y1 and y2 form a fundamental set of...

Determine which of the following pairs of functions y1 and y2 form a fundamental set of solutions to the differential equation: x^2*y'' - 4xy' + 6y = 0 on the interval (0, ∞). Mark all correct solutions:

a) y1 = x and y2 = x^2

b) y1 = x^2 and y2 = 4x^2

c) y1 = x^2 and y2 = x^3

d) y1 = 4x^2 and y2 = x^3

e) y1 = [(x^2)+(x^3)] and y2= x^3

Expert Solution

Colby Messinger answered 1 year ago

a) verify that y1 and y2 are fundamental solutions b) find the general solution for the...

a) verify that y1 and y2 are fundamental solutions b) find the general solution for the given differential equation c) find a particular solution that satisfies the specified initial conditions for the given differential equation 1. y'' + y' = 0; y1 = 1 y2 = e^-x; y(0) = -2 y'(0) = 8 2. x^2y'' - xy' + y = 0; y1 = x y2 = xlnx; y(1) = 7 y'(1) = 2

Find two integer pairs of the form (x,y) with |x|<1000 such that 22x+28y=gcd(22,28) (x1,y1)=( (x2,y2)=(

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential...

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential equation b) find the general solution for the given differential equation c) find a particular solution that satisfies the specified initial conditions for the given differential equation y'' - y = 0 y1 = e^x, y2 = e^-x : y(0) = 0, y'(0) = 5

a) The functions y1 = x^2 and y2 = x^5 are two solutions of the equation...

a) The functions y1 = x^2 and y2 = x^5 are two solutions of the equation x^2 y ″ − 6 x y ′ + 10 y = 0. Let y be the solution of the equation x^2 y ″ − 6 x y ′ + 10 y = 3 x^5 satisfyng the conditions y ( 1 ) = 0 and y ′ ( 1 ) = 1. Find the value of the function f ( x ) = y ( x )...

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(1 point) In general for a non-homogeneous problem y′′+p(x)y′+q(x)y=f(x) assume that y1,y2 is a fundamental set...

(1 point) In general for a non-homogeneous problem y′′+p(x)y′+q(x)y=f(x) assume that y1,y2 is a fundamental set of solutions for the homogeneous problem y′′+p(x)y′+q(x)y=0. Then the formula for the particular solution using the method of variation of parameters is yp=y1u1+y2u2 where u′1=−y2(x)f(x)W(x) and u′2=y1(x)f(x)W(x) where W(x) is the Wronskian given by the determinant W(x)=∣∣∣y1(x)y′1(x)y2(x)y′2(x)∣∣∣ So we have u1=∫−y2(x)f(x)W(x)dx and u2=∫y1(x)f(x)W(x)dx. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. In other words we have...

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