Ordinary Differential Equations/Substitution 4

1)

$y'=csc(x+y)-1$

$v=x+y$

$v'=1+y'$

$v'-1=csc(v)-1$

$sin(v)dv=dx$

$\int sin(v)dv=\int dx$

$-cos(v)=x+C$

$-cos(x+y)=x+C$

$y=arccos(-x+C)-x$

2)

$y'=csc({\frac {y}{x}})+{\frac {y}{x}}$

$v={\frac {y}{x}}$

$v'x+v=y'$

$v'x+v=csc(v)+v$

$v'x=csc(v)$

$sin(v)dv={\frac {dx}{x}}$

$\int sin(v)dv=\int {\frac {dx}{x}}$

$-cos(v)=ln(x)+C$

$-cos({\frac {y}{x}})=ln(x)+C$

$y=xarccos(-ln(x)+C)$

3)

$ycos(y^{2})y'-sin(y^{2})=0$

$v=sin(y^{2})$

$v'=2yy'cos(y^{2})$

${\frac {v'}{2}}-v=0$

$v'=2v$

${\frac {dv}{v}}=2dx$

$\int {\frac {dv}{v}}=\int 2dx$

$ln(v)=2x+C$

$v=Ce^{2x}$

$sin(y^{2})=Ce^{2x}$

$y^{2}=arcsin(Ce^{2x})$

4)

$y'=yln(y)+y$

$v=ln(y)$

$v'={\frac {y'}{y}}$

$v'y=yv+y$

$v'=v+1$

${\frac {dv}{v+1}}=dx$

$\int {\frac {dv}{v+1}}=\int dx$

$ln(v+1)=x+C$

$v+1=Ce^{x}$

$v=Ce^{x}-1$

$ln(y)=Ce^{x}-1$

$y=e^{Ce^{x}-1}$

5)

$y'=(x^{2}+y-1)^{2}-2x$

$v=x^{2}+y-1$

$v'=2x+y'$

$2x+y'=(x^{2}+y-1)^{2}$

$v'=v^{2}$

${\frac {dv}{v^{2}}}=dx$

$\int {\frac {dv}{v^{2}}}=\int dx$

$-{\frac {1}{v}}=x+C$

$v={\frac {-1}{x+C}}$

$x^{2}+y-1={\frac {-1}{x+C}}$

$y={\frac {-1}{x+C}}-x^{2}+1$

6)

$y'={\frac {x^{2}}{y^{2}}}+{\frac {y}{x}}$

$v={\frac {y}{x}}$

$y'=v+xv'$

$v+xv'={\frac {1}{v^{2}}}+v$

$v^{2}dv={\frac {dx}{x}}$

$\int v^{2}dv=\int {\frac {dx}{x}}$

${\frac {1}{3}}v^{3}=ln(x)+C$

${\frac {y^{3}}{3x^{3}}}=ln(x)+C$

$y=(3x^{3}(ln(x)+C))^{\frac {1}{3}}$