Question Show that y1=e^(−2x) is a solution to the differential equation (2x+1)y′′+2(4x+1)y′+8xy=0, and use reduction of order to find the solution corresponding to the initial data y(0)=1 , y′(0)=3 .
Show that y1=e^(−2x) is a solution to the differential equation (2x+1)y′′+2(4x+1)y′+8xy=0, and use reduction of order to find the solution corresponding to the initial data y(0)=1 , y′(0)=3 .
【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/2The differential equation is of second order. And one of its solution is already provided. Given (2x+1)y^('')+2(4x+1)y^(')+8xy=0.-(1) and a solution y_(1)=e^(-2x)In order to show y_(1)=e^(-2x) is a solutiony_(1)^(')=-2e^(-2x),y_(1)^('')=4e^(-2x)divide equ(1) through by 2x+1y^('')+(2(4x+1)y^('))/(2x+1)+(8xy)/(2x+1)=0Comparing with the general formy^('')+P(x)y^(')+Q(x)y=0Ther, P(x)=(2(4x+1))/(2x+1),Q(x)=(8x)/(2x+1)ExplanationKindly check through the attached image very well for proper explanations.Explanation:Please refer to solution in this step.Step2/2The tw ... See the full answer