Answer to Question #158270 in Differential Equations for Randal Rodriguez

Question #158270

find the general and particular solution of dy/dx=-8sin4x ; y=6 when x=0

find the general and particular solution of dy/dx=3y ; y=3 when x=0


1
Expert's answer
2021-01-26T11:51:44-0500

Solution

First equation can be integrated directly. General solution is

"y = -8\\int{sin(4x) dx}+C = 2cos(4x)+C"

C is arbitrary constant.

From initial condition y=6 when x=0 we’ll get 2+C=6   =>   C = 4   =>    particular solution is y = 2cos(4x)+4

Second equation may be solved as equation with separable variables

dy/y = 3dx  =>  ln(y) = 3x+c  => y = e3x+c = Ce3x

The same result for general solution can be obtained for this equation as linear equation with constant coefficients.

From initial condition y=3 when x=0 we’ll get Ce0=3   =>   C = 3   =>    particular solution is y = 3e3x  

Answer

First equation. General solution is y = 2cos(4x)+4, particular solution is y = 2cos(4x)+4.

Second equation. General solution is y = Ce3x , particular solution is y = 3e3x .


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