Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

My problem is this given Cauchy-Euler equation:

$$x^{3}y^{\prime \prime \prime} +xy^{\prime}-y=0$$

My approach: this is an differential equation, so i was looking for a solution with the method of undetermined coefficients. but honestly, i failed.

P.S. edits were made to improve language and latex

share|cite|improve this question

2 Answers 2

up vote 4 down vote accepted

Besides to another answer, you may substitute $y=x^m$ wherein $m$ is a number and then form the certain $y_c(x)$. Let's do that: $$y=x^m\to y'=mx^{m-1},~~y'''=m(m-1)(m-2)x^{m-3}$$ So $$x^3y'''+xy'-y=0\Longrightarrow x^m(m(m-1)(m-2)+m-1)=0$$ If $x\neq 0$ then $$(m-1)^3=0$$ This means that $$y_c(x)=x^1(1+\ln x+\ln^2 x), ~x>0$$

share|cite|improve this answer
$\quad \langle +\rangle_+^+\quad \bf \ddot\smile\;$ – amWhy Jun 16 '13 at 1:07

Hint: Use substitute $t=\log x$ for $x>0$ and $t=\log (-x)$ for $x<0$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.