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)

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

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

$$v'x+v=y'$$

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

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

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

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

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

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

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

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)

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

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

$$y'=v+xv'$$

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

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

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

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

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

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