This problem will walk you through finding the general solution to the homogeneous linear equation

$\displaystyle {x}{y}{''}+{2}{y}'={0}$

First, reduce the equation to a first-order equation using the substitution $\displaystyle {w}={y}'$. Solve the first-order equation using separation of variables, integrate, and write the solution without coefficients or constants of integration.

$\displaystyle {y}_{{1}}=$ Preview Question 6 Part 1 of 3  

Now use the solution $\displaystyle {y}_{{1}}$ you found and the reduction-of-order substitution $\displaystyle {y}={u}{y}_{{1}}$ to find a non-trivial solution to the original equation. Write the solution without coefficients or constants of integration.

$\displaystyle {y}_{{2}}=$ Preview Question 6 Part 2 of 3  

Finally, write the general solution in the form $\displaystyle {y}={c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}$. Use $\displaystyle {c}_{{1}}$ and $\displaystyle {c}_{{2}}$ as your constants.

$\displaystyle {y}=$ Preview Question 6 Part 3 of 3