Integrate : dy/dx = 2x + y ?

2 Answers
Apr 1, 2018

x^2+xy+c

Explanation:

=intdy=int2x+ydx

=y=(2x^2)/2+xy+c

=y=x^2+xy+c

Hope it helps!

Apr 1, 2018

" The GS is, "y*e^-x+2(x+1)e^-x=C, or,

y+2(x+1)=Ce^x.

Explanation:

Rewriting the given diff. eqn. (DE) as dy/dx-y=2x, we find

that it is a linear DE of the form : dy/dx+yP(x)=q(x).

To find its gen. soln. (GS), we need to multiply it by the

integrating factor (IF) e^(intP(x)dx.

Since,

P(x)=-1, intP(x)dx=int-1dx=-x :." IF is "e^-x.

Multiplying the DE by IF, we get,

e^-xdy/dx-ye^-x=2xe^-x.

:. e^-x*d/dx(y)+y*d/dx(e^-x)=2xe^-x, or,

d/dx(y*e^-x)=2xe^-x.

:. y*e^-x=int2xe^-xdx+C,

=2[x*inte^-xdx-int{d/dx(x)inte^-xdx}dx]+C......[because," Integration by Parts]",

=2[x(-e^-x)-int(-e^-x)dx]+C,

=2[-xe^-x-e^-x]+C.

rArr" The GS is, "y*e^-x+2(x+1)e^-x=C, or,

y+2(x+1)=Ce^x.

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