How to solve this ordinary differential equation \(\displaystyle{t}{x}{'''}+{3}{x}{''}-{t}{x}'-{x}={0}\)? For

nastupnat0hh

nastupnat0hh

Answered question

2022-04-01

How to solve this ordinary differential equation
tx+3xtxx=0?
For equation tx+3xtxx=0, we know a special solution x1=1t, how to general solution?
I firstly attempted d(tx+2xtx)=0, then tx2xtx=C, C is a constant.But in next step , I found that my solution is wrong.Since x1=1t is a special solution of tx+3xtxx=0, we found x2=1t is a solution of equation.Then x=x1x2=2t is a solution of tx+2xtx=0. As you can see ,the step is wrong.
Then I attemped other way to solve this equation ,but all failed.Could help me solve this equation?

Answer & Explanation

Regan Gallegos

Regan Gallegos

Beginner2022-04-02Added 9 answers

Step 1
Use Reduction of Order, we have
tx+3xtxx=0,≈≈x1=d1t
Let x2=vx1=dvt
Step 2
Taking derivatives, substituting and simplifying into tx2+3x2tx2x2=0
vv=0
Let w=v, so we have
ww=0w=aet+bet
This gives v=aet+bet+c
From the initial substitution x2=dvt=daet+bet+ct.

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