Verify that the indicated function is a solution of the give

Verify that the indicated function is a solution of the given differential equation. Assume the appropriate interval I of definition of each solution. a) dy/dx + y = x + 1 y = x + 3e^-x b) d^2y/dx^2 - 2 dy/dx - 8y = 0 y = c_1e^4x + c_2e^-2x

Solution

a)

Differentiating given function gives

y\'=1-3e^{-x}=1+x-x-3e^{-x}=1+x-(x+3e^{-x})=1+x-y

So,y\'+y=1+x

Hence given function is solution to the ode

b)

Differentiating given function gives

y\'=4c1 e^{4x}-2c2 e^{-2x}

y\'\'=16c1 e^{4x}+4c2 e^{-2x}

So,

y\'\'-2y\'-8y=16c1 e^{4x}+4c2 e^{-2x}-2(4c1 e^{4x}-2c2 e^{-2x})-8(c1 e^{4x}+c2 e^{-2x})

               =(16c1-8c1-8c1)e^{4x}+(4c2+4c2-8c2)e^{-2x}=0

Hence , given function satisfies the differential equation.