Find the general solution to the equation x(dy/dx)+3(y+x^2)=(sin x)/x

UkusakazaL

UkusakazaL

Answered question

2021-02-20

Find the general solution to the equation x(dydx)+3(y+x2)=sinxx

Answer & Explanation

Lacey-May Snyder

Lacey-May Snyder

Skilled2021-02-21Added 88 answers

Write this equation as follows:
dydx+3yx+sinxx23x(1)
We first solve the differential equation
dydx+3yx=0dyy=3dxxdyy=3dxx
Thus,
ln|y|=3ln|x|+C=ln1|x3|+C
Here we used that αlnβ=Inβα. Now use the exponential function:
|y|=1|x3|eC
Recall that |α|=±α, so we can write
y=±eC1x3
Defining a new constant D=±eC yields
y=Dx3
This is the solution of the homogeneous equation. To obtain the solution of the initial equation, we regard D as a function of x, so
dydx=Dx33Dx2x6=Dx33Dx4
Plugging this and (2) into (1) yields
Dx33Dx4+3Dx4=sinxx23xDx3=sinxx23x
Thus,
D=xsinx3x4D=(xsinx3x4)dx=xsinxdx3x4dx
Now,
xsinxdx={(u=xdu=dx),(dv=sinxdxv=cosx)}
=xcosx+cosxdx
=xcosx+sinx+C
Therefore,

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