solve
dy/dx + (y logy)/x = y(logy)^2/(x^2)
Let me rephrase: you have to translate the differential equation from having y's in it to having only u's in it. So,
$\displaystyle u(x)=\ln(y(x))$ implies that
$\displaystyle e^{u(x)}=y(x),$ and
$\displaystyle u'(x)=\dfrac{y'(x)}{y(x)},$ and hence
$\displaystyle e^{u(x)}u'(x)=y'(x).$
Now plug this information into the differential equation. What do you get?
If you have studied Bernoulli equations, then surely you have studied simple substitutions, since solving a Bernoulli equation involves a substitution! That is all we are doing here. The result of plugging in the u expressions into the differential equation is going to be a Bernoulli equation. You are being asked to solve a multi-step problem. I am not going to provide you with the next step. You need to plug in the expressions I have given you into the original differential equation. What do you get when you do that?
I understand a problem given in the text book if there is steps. But when given a different problem , i wont be able to do it.
May be iam stupid.And also When doing i a problem i always make stupid mistakes...
Ok. So let me ask you this question: how badly do you want to be able to solve different problems, that are not exactly like problems in the text? How would you like to be able to solve problems the like of which you have never seen before?
How badly do you want to avoid "stupid mistakes"?
If you are even taking a differential equations course, then I would say you are not stupid. Every teacher who has ever lived knows that there are faster students and slower students. Maybe you are one of the slower students. I certainly was. I had to re-take Algebra I and II both, while I was in high school. And then I took Calculus. And then I re-took Calculus. And then I re-took Calculus. I had to take Calculus 3 times, and even then I did not really understand the concept of the limit. I did not understand limits until after I had taken Calculus III (Multivariable), Complex Variables, and then finally Classical Analysis (introductory real analysis). Now, I would say, I know my basics pretty well, and that has stood me in good stead.
Being a slower student is not a bad thing. Perhaps that means you absorb material more deeply, and you digest it more than others.
I say these things to encourage you. I am not going to sit here and say that differential equations are easy. I did not find it easy. But that does not mean that you can not learn differential equations.
So, getting back to my earlier questions, here is the summary question: how badly do you want to learn this stuff?