Solve differential equation xu'(x)= u^2-4

Solve differential equation xu'(x)= u^2-4

Question
Solve differential equation \(xu'(x)= u^2-4\)

Answers (1)

2021-03-08
\(xu'(x)= u^2-4\)
\(xu'(x) 1/(u^2-4)=1\)
\(1/(u^2-4) du=1/x dx\)
Integrating on both sides
\(int 1/(u^2-4)du= int 1/x dx\)
The partial fraction of the left side \(1/((u+2)(u−2))= −1/(4(x+2))+1/(4(x−2))\)
\(-1/4 int 1/(u+2)du +1/4 int 1/(u-2)du= int 1/x dx\)
\(-1/4 ln (u+2)du+1/4 ln(u-2)= ln(x)+C_1\)
\(-ln(u+2)du+ln(u-2)= 4 ln(x)+4C_1\)
\(ln((u-2)/(u+2))= ln(x^4)+C_2\)
\(((u-2)/(u+2))= (x^4)+e^(C_2)\)
\(= x^4+C\)
\(u= - (2(C+x^4))/(-C+x^4)\)
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