Question

Integrate : dy/dx = 2x + y ?

Answer 1

`x^2+xy+c`

Explanation:

=`intdy=int2x+ydx`

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

=`y=x^2+xy+c`

Hope it helps!

Answer 2

`" 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`.

Feel the Joy of Maths.!