Find all values of m the for which the function y=emx is a solution of the given differential equation. (NOTE : If there is more than one value for m write the answers in a comma separated list.) (1) y′′+3y′-4y=0, (2) y′′′+2y′′-3y′=0

1) m=[1,4]

2) m=[-3,0,1]

Step-by-step explanation:

for y = e^ (m*x), then

y′=m*e^ (m*x)

y′′=m²*e^ (m*x)

y′′′=m³*e^ (m*x)

thus

1) y′′+3y′-4y=0

m²*e^ (m*x) + 3*m*e^ (m*x) - 4*e^ (m*x) = 0

e^ (m*x) * (m²+3*m-4) = 0 → m²+3*m-4 = 0

m = [-3±√ (9-4*1 * (-4) ] / 2 → m₁=-4, m₂=1

thus m=[1,4]

2) y′′′+2y′′-3y′=0

m³*e^ (m*x) + 2*m²*e^ (m*x) - 3*m*e^ (m*x) = 0

e^ (m*x) * (m³+2*m²-3m) = 0 → m³+2*m²-3m=0

m³+2*m²-3m = m * (m²+2*m-3) = 0

m=0

or

m = [-2±√ (4-4*1 * (-3) ] / 2 → m₁=-3, m₂=1

thus m=[-3,0,1]