Question-Bernoulli

[mathmari]

Hi! I have a question..

Given the Bernoulli equation:
\(\displaystyle 2xyy'+(1+x)y^2=e^{2x} \), \(\displaystyle x>0 \)

\(\displaystyle lim_{x -> 0^{+}} y(x) <\infty \)


The transformation is \(\displaystyle u=y^{2} \).

So, \(\displaystyle u'+(\frac{1}{x}+1)u=\frac{e^{2x}}{x}\).

How can I find the initial value \(\displaystyle u(1)\) so that \(\displaystyle lim_{x -> 0^{+}} u(x) <\infty \) ??

 

[Deleted member 4993]

Hi! I have a question..

Given the Bernoulli equation:
\(\displaystyle 2xyy'+(1+x)y^2=e^{2x} \), \(\displaystyle x>0 \)

\(\displaystyle lim_{x -> 0^{+}} y(x) <\infty \)


The transformation is \(\displaystyle u=y^{2} \).

So, \(\displaystyle u'+(\frac{1}{x}+1)u=\frac{e^{2x}}{x}\).

How can I find the initial value \(\displaystyle u(1)\) so that \(\displaystyle lim_{x -> 0^{+}} u(x) <\infty \) ??

What was the initial value for your original ODE?

 

[mathmari]

The solution of this ODE is \(\displaystyle u(x)=\frac{3ce^{-x}+xe^{3x}}{3x}\).

How can I find the value of c, so that \(\displaystyle lim_{x \rightarrow 0^{+}} u(x)< \infty\)??