Differential Equations Questions

Question

1. Write and then solve for the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y²."

 

2. Solve the differential equation dy/dx = [ycos(x)]/1+y² with the initial condition y(0) = 1. 

Answer 1

1.y'=1/y²

y²dy=dx

y³/3+C=x 

 

2. dy/dx = [ycos(x)]/1+y²

((1+y²)/y)dy=cos(x)dx

ln(y)+y²/2=sin(x)+C

y(0)=1

ln(1)+1/2=sin0+C, C=1/2

ln(y)+y²/2=sin(x)+1/2