1} Use the following steps to solve the following IVP y \'\' + 6 y \' + 8 y = 3

Question

1} Use the following steps to solve the following IVPy''+ 6y'+ 8y= 3e^-x: y(0) = 0,,,y'(0) =1
,,,,,,,,,# Assume that y(x)=Yh(x)+Yp(x) ,,,,,,,,#Find the solution of Yh(x) which is the solution of the homogeneous equation y''+6y'+8y=0 ,,,,,,,# use the method of undetermined coefficients to find Yp(x) ,,,,,,_ use the table to find the form od Yp(x) . In this case Yp(x)=Ae^-x ,,,,,,_check if Yp(x) is one of the basis of the solution of Yh(x).If so use the modification rul.Check again. ,,,,,,,,_Substitute Yp(x) into ODE to find the constant of Yp(x).i.e find A ,,,,,,,,_Find the particular solution of the ODE by using the initial conditions in Y(x)=Yh(x)+Yp(x)

,,,,,,2} solve the following ODE;
,,,,,,,,a) y''+4y'+3y=4e^-x. ,,,,,,,y(0)=0,,,,,,,,y'(0)=1 ( use modification rule) ,,,,,,,b) y''+4y'+3y=9x,,,,,,, find the general solution ,,,,,,c)y''+4y'+3y=4e^-3x+9x,,,,,y(0)=1,,,y'(0)=10 ( use summation rule)
,,,,,,3} find the solution of the following ODE; ,,,,,,y''+4y'+8y=0,,,,y(0)=0,,,,,y'(0)=10 1} Use the following steps to solve the following IVPy''+ 6y'+ 8y= 3e^-x: y(0) = 0,,,y'(0) =1
,,,,,,,,,# Assume that y(x)=Yh(x)+Yp(x) ,,,,,,,,#Find the solution of Yh(x) which is the solution of the homogeneous equation y''+6y'+8y=0 ,,,,,,,# use the method of undetermined coefficients to find Yp(x) ,,,,,,_ use the table to find the form od Yp(x) . In this case Yp(x)=Ae^-x ,,,,,,_check if Yp(x) is one of the basis of the solution of Yh(x).If so use the modification rul.Check again. ,,,,,,,,_Substitute Yp(x) into ODE to find the constant of Yp(x).i.e find A ,,,,,,,,_Find the particular solution of the ODE by using the initial conditions in Y(x)=Yh(x)+Yp(x) 1} Use the following steps to solve the following IVPy''+ 6y'+ 8y= 3e^-x: y(0) = 0,,,y'(0) =1
,,,,,,,,,# Assume that y(x)=Yh(x)+Yp(x) ,,,,,,,,#Find the solution of Yh(x) which is the solution of the homogeneous equation y''+6y'+8y=0 ,,,,,,,# use the method of undetermined coefficients to find Yp(x) ,,,,,,_ use the table to find the form od Yp(x) . In this case Yp(x)=Ae^-x ,,,,,,_check if Yp(x) is one of the basis of the solution of Yh(x).If so use the modification rul.Check again. ,,,,,,,,_Substitute Yp(x) into ODE to find the constant of Yp(x).i.e find A ,,,,,,,,_Find the particular solution of the ODE by using the initial conditions in Y(x)=Yh(x)+Yp(x)

,,,,,,2} solve the following ODE;
,,,,,,,,a) y''+4y'+3y=4e^-x. ,,,,,,,y(0)=0,,,,,,,,y'(0)=1 ( use modification rule) ,,,,,,,b) y''+4y'+3y=9x,,,,,,, find the general solution ,,,,,,c)y''+4y'+3y=4e^-3x+9x,,,,,y(0)=1,,,y'(0)=10 ( use summation rule)
,,,,,,3} find the solution of the following ODE; ,,,,,,y''+4y'+8y=0,,,,y(0)=0,,,,,y'(0)=10

Explanation / Answer

1)y''+ 6y'+ 8y= 3e^-x: y(0) = 0,,,y'(0) =1

m^2+6m+8=0

m= -4 and -2

yh=c1e^-4x+c2e^-2x

using method of undetermined coefficints

if we diff 3e^-x repeatedly we get Ae^-x repeatedly

so yp=Ae^-x

substitute it in DE

we get

A=1

so y=yh+yp=c1e^-4x+c2e^-2x +e^-x

now substitute

y(0) = 0=c1+c2+1,,,y'(0) =1=-4c1-2c2-1

c2=-1

c1=0

2)a) y''+4y'+3y=4e^-x. ,,,,,,,y(0)=0,,,,,,,,y'(0)=1

m^2+4m+3=0

m=-3,-1

yh=c1e^-3x+c2e^-x

yp=Ae^-x

now substitute yp in DE and get A

y=c1e^-3x+c2e^-x +Ae^-x

now substitutey(0)=0,,,,,,,,y'(0)=1 and get c1 and c2

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