Solve the initial value problem correctly

Question

with

a) To solve this, we should use the substitution

  

 

Enter derivatives using prime notation (e.g., you would enter  for ).

b) After the substitution from the previous part, we obtain the following linear differential equation in .

c) The solution to the original initial value problem is described by the following equation in .

Answer 1

a) u² = x² + y²; 2uu' = 2x + 2yy'; x + yy' = uu'.

And we get equation: uu' = u, or du/dx =1

b) We integrate this equation: du = dx; ∫du = ∫dx; u = x + C;

√(x² + y²) = x + C; 4 = 2 + C; C = 2

c) So, solution of IVP in implicite form: √x² + y² = x + 2; y² = 4x + 4