What is the general solution of the differential equation # dy/dx +3y = e^(4x) #?

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Answer 1 · 2017-10-06T23:43:45

# y = 1/7e^(4x) + Ce^(-3x) #

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

# dy/dx + P(x)y=Q(x) #

Ther equation is already in standard form:

# dy/dx +3y = e^(4x) # ..... [A}

Then the integrating factor is given by;

# I = e^(int P(x) dx) #
# \ \ = exp(int \ 3 \ dx) #
# \ \ = exp( 3x ) #
# \ \ = e^(3x) #

And if we multiply the DE [A] by this Integrating Factor, #I#, we will have a perfect product differential;

# dy/dx +3y = e^(4x) #

# :. e^(3x)dy/dx +3e^(3x)y = e^(3x)e^(4x) #

# :. d/dx( e^(3x)y) = e^(7x) #

Which we can directly integrate to get:

# \ \ \ \ \ e^(3x)y = 1/7e^(7x) + C #

# :. y = 1/7e^(7x)e^(-3x) + Ce^(-3x) #

# :. y = 1/7e^(4x) + Ce^(-3x) #