Question

Use an integrating factor to solve the following.

1. xdy - xdx + ydx =
2. 4y - dy/dx = xe^2x

Answer 1

1. Incorrect question.

x dy - xdx + ydx = ?

2. Check the attached image.

Answer 2

Answer

1. X/Y-x^2/2

2. X=e^4x/8

explaintion

1(.xdy+ydx)-xdx

get intragation sign we have

[integration(xdy+ydx)]-integration (xdx)

= X/Y-x^2/2

because [integration(udv+vdu)=u/v]

2.mutliplying by (-)

dy/dx-4y=xe^2x

now form is dy/dx-Py=Q

where P=4 and Q=xe^2x

I.F(introgative factor)=integration(e^pdx)

=integration(e^4dx)

= integration (e^4x)

now

x×I.F=Integration{(Q×I.F)dx}

x×e^4x=integration (e^2x×e^4x)dx

x×e^4x=e^8x/8

X=e^4x/8