This is a homework that I'm having a bit of trouble with:

Find a general solution of:


Of course this should be done using the method of characteristics but I'm having trouble solving the characteristic equations since none of the equations decouple:


Any suggestions?

  • $\begingroup$ Because people have already answered the question you asked, you should simply start a new question with the correction. Otherwise, the answers that were already given for the question you asked will no longer appear to match the question. I edited this question to restore the original wording. It's perfectly fine to ask a second question, there is no limitation of resources or anything like that to worry about. $\endgroup$ – doraemonpaul Jan 24 '13 at 1:53

Follow the method in http://en.wikipedia.org/wiki/Characteristic_equations#Example:




$\dfrac{du}{dt}=2$ , letting $u(0)=0$ , we have $u=2t$

$\dfrac{dy}{dt}=2y$ , letting $y(0)=y_0$ , we have $y=y_0e^{2t}=y_0e^u$



Let $w=x^2$ ,

Then $\dfrac{dw}{dt}=2x\dfrac{dx}{dt}$




I.F. $=e^{\int\frac{1}{t}dt}=e^{\ln t}=t$







  • $\begingroup$ How to handle $y_0$ ? Treat it as a constant? Or what? Can somebody help me? $\endgroup$ – doraemonpaul Jan 20 '13 at 21:26
  • $\begingroup$ Actually there was a typo in the original post. It has been updated $\endgroup$ – rmh52 Jan 23 '13 at 19:33
  • $\begingroup$ Thank you about the hint in math.stackexchange.com/questions/285384 given by @Fabian, in fact it just treat $y_0$ as a constant. $\endgroup$ – doraemonpaul Jan 26 '13 at 0:58

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