Differential Equations Exercise Questions

Solve the following equations by reduction of order.

a$.x{y}^{\text{'}\text{'}}=(y^{\text{'}}{)}^2$;

b$.2x{y}^{\text{'}}{y}^{\text{'}\text{'}}=(y^{\text{'}}{)}^2-1$;

c$.y^3{y}^{\text{'}\text{'}}=1$;

d$.y{y}^{\text{'}\text{'}}+1=(y^{\text{'}}{)}^2$;

e$.y^{\text{'}\text{'}}(e^x+1)+y^{\text{'}}=0$;

f$.y^{\text{'}\text{'}}=e^y$;

g$.y{y}^{\text{'}\text{'}}+(y^{\text{'}}{)}^2=1$;

h$.y^{\text{'}\text{'}\text{'}}=2y^{\text{'}\text{'}}$;

i$.y^{\text{'}\text{'}\text{'}}+3y^{\text{'}}{y}^{\text{'}\text{'}}=0,$

j$.y^{\text{'}}{y}^{\text{'}\text{'}\text{'}}=2(y^{\text{'}\text{'}}{)}^2.$