Answer to Question #158270 in Differential Equations for Randal Rodriguez

Question

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

Answer 1

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 = e^3x+c = Ce^3x

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 = 3e^3x  

Answer

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

Second equation. General solution is y = Ce^3x , particular solution is y = 3e^3x .