Question

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

Answer 1

1) m=[1,4]

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

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]