Solve this cauchy differential equation:
x^2y'' + xy' + y = (lnx)^2+2x

For the corresponding homogeneous equation,

we can look for a solution of the form , with derivatives and . Substituting these into the ODE gives

which admits two solutions, and , which we can write as

and by the same token,

so we see two independent solutions that make up the characteristic solution,

For the non-homogeneous ODE, we make the substitution

so that by the chain rule, the first derivative becomes

Let . Then the second derivative becomes

Substituting these into the ODE gives

Look for a particular solution , which has second derivative . Substituting these into the ODE gives

so that the particular solution is

Solving in terms of gives the solution

and the overall general solution is

let soccer = x and golf = y

part a

1st equation: x + y = 125

2nd equation: x = y +45

part b

replace x in the 1st equation with the 2nd one:

y + 45 + y = 125

simplify:

2y + 45 = 125

subtract 45 from each side:

2y = 80

divide both sides by 2:

y = 80/2

y = 40

she plays golf for 40 minutes.

part c

no because she plays soccer for 45 minutes longer than golf 40 +45 =   85 minutes, not 80 minutes.