solving second order linear ODEaxy′(x)+12y″(x)=0where a is a constant. Also how can I solve this...

Tara Alvarado

Tara Alvarado

Answered

2022-01-20

solving second order linear ODE
axy(x)+12y(x)=0
where a is a constant. Also how can I solve this equation if the right hand side is different than zero?

Answer & Explanation

Papilys3q

Papilys3q

Expert

2022-01-20Added 34 answers

Step 1
Consider z=y. Then this is just axz(x)+12z(x)=0. That is z=2axz which is separable.
Step 2
ln|z|=dzz=2axdx=2aln|x|+C0 and so y=z=C1x2a.
Integrating yields y=C112ax12a+C2
Karen Robbins

Karen Robbins

Expert

2022-01-21Added 49 answers

Step 1
Assuming a solution in an interval not containing x=0, lets rewrite the equation as:
2ay+xy=0
Now put y=v,y=v to get
2av+xv=0
Step 2
Separating variables produces
2ax=vv
2alogCx=logv
2alogCx=logv
Kx2a=y
Kx12a12a+C=y and a12
RizerMix

RizerMix

Expert

2022-01-27Added 437 answers

Substitute: \(y=u

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