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#.

Feel the Joy of Maths.!