How do you find the general solution to dy/dx = -2y , y=3 when x=0?

2 Answers
Apr 22, 2018

# y = 3 e^(- 2x)#

Explanation:

This is separable:

#(dy)/y = - 2 \ dx#

#implies ln |y| = - 2x + C#

# |y| = C e^(- 2x)#

#y_o = 3 implies C = 3#

and:

# y = 3 e^(- 2x)#

Apr 22, 2018

General solution is when the constants of integration are not evaluated

#y=Ae^(-2x)#

Particular solution is when the constant of integration is evaluated

#y=3e^(-2x)#

Explanation:

#(dy)/(dx)=-2y#

and

#y=3" when " x=0#

#(dy)/(dx)=-2y#

separate variables and integrate

#int(dy)/y=-int2dx#

#=>lny+c=-2x#

let #lc=lnk#

we have therefore

#lny+lnk=-2x#

#=>lnky=-2x#

#=>ky=e^(-2x)#

#y=1/ke^(-2x)#

#A=1/k#

#:.y=Ae^(-2x)--(1)#

this is the General solution

using the boundary conditions

#y=3" when " x=0#

#3=Ae^0#

#:.A=3#

Particular solution

#y=3e^(-2x)#