Double checking needed on 2 Differential Equations

[VinnyCee]

Problem 1:

Find solution for:

$$(x^2\,-\,4)\,y''\,+\,(3\,x)\,y'\,+\,y\,=\,0$$

using power series methods.

Answer 1:

I get a recursion formula:

$$a_{n\,+\,2}\,=\,\frac{n\,+\,1}{4\,(n\,+\,2)}\,a_n$$

and a final answer:

$$y(x)\,=\,a_0\,\left[1\,+\,\frac{x^2}{8}\,+\,\frac{3}{128}\,x^4\,+\,\frac{5}{1024}\,x^6\,+\,...\right]\,+\,a_1\,\left[x\,+\,\frac{x^3}{6}\,+\,\frac{x^5}{30}\,+\,\frac{x^7}{140}\,+\,...\right]$$

Does that look right?



Problem 2:

Use Euler's method to solve:

$$(2\,x^2)\,y''\,+\,(x)\,y'\,+\,y\,=\,0$$

Answer 2:

Using the quadratic equation to solve for r:

$$2\,r^2\,-\,r\,+\,1\,=\,0$$

$$r\,=\,\frac{1}{4}\,\pm\,\frac{\sqrt{7}}{4}\,i$$

Which means that:

$$\lambda\,=\,\frac{1}{4}$$ AND $$\mu\,=\,\frac{\sqrt{7}}{4}$$

And finally:

$$y(x)\,=\,C_1\,x^{\frac{1}{4}}\,cos\,(\frac{\sqrt{7}}{4}\,ln\,x)\,+\,C_2\,x^{\frac{1}{4}}\,sin\,(\frac{\sqrt{7}}{4}\,ln\,x)$$

Thanks for the checking in advance!

 

[Justin Lazear]

Mathematica agrees with both of your solutions. Well done.

--J

 

[VinnyCee]

Thanks alot

I need to get that program someday!

 

[Justin Lazear]

It's possible that your college has a license for it and will give it to you. Why don't you contact your IT department and ask?

--J