Solve the linear equations by considering y as a function of x, that is, y = y(x) 3xy'+y=12x

Amari Flowers 2021-02-12 Answered
Solve the linear equations by considering y as a function of x, that is,
y=y(x)
3xy+y=12x
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Expert Answer

SchepperJ
Answered 2021-02-13 Author has 96 answers

Variation of parameters
First, solve the linear homogeneous equation by separating variables. Rearranging terms in the equation gives
3xy=ydydx=y3xdyy=dx3x
Now, the variables are separated, x appears only on the right side, and y only on the left.
Integrate the left side in relation to y, and the right side in relation to x
dyy=13dxx
Which is
ln|y|=13ln|x|+c
By taking exponents, we obtain
|y|=e13ln|x|+c=|x|13ec
Hence,we obtain
y=Cx13
where C=±ecandyc=x13 is the complementary solution .
Next, we need to find the particular solution yp
Therefore, we consider uyc, and try to find u, a function of x, that will make this work.
Let's assume that uyc is a solution of the given equation. Hence, it satisfies the given equation.
Write the equation in the standard form (divide iy by  3x0)
y+y3x=12
Substituting uyc, and its derivative in the equation gives
(uyc)+uyc3x=4
uyc+uyc+uyc3x=4
uyc+u(yc+yc3x)=0  =4
since yc is a solution
Therefore,
uyc=4u=4yc
which gives
u=4ycdx
Now, we can find the function u:
u=4x13dx
=4x13dx
=4x13+113+1+c
=4x4343+c
=434x43+c

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