Consider the equation
xy'' - y' + 4x^(3)y = 0
show that y1(x) = sin(x^2) is a solution
find the second solution by letting y = u(x)y1 (x) and solve for u(x)
write the complete general solution

Substituting and its derivatives into the ODE gives

as required.

Assume a second solution of the form . Then

Substitute these into the ODE:

This ODE is separable as

Integrating both sides gives

For the remaining integral, we have

So we have

Integrate both sides to solve for :

Then our second solution turns out to be

already captures the second term here, so the second fundamental solution is

and the general solution is

3: 2

step-by-step explanation:

shorter piece is 2: 3, so just reverse it, and you get 3: 2

0.16 or 16%

step-by-step explanation:

lets take a coin for example.

if you toss it twice, it has a 1/4 chance of landing on heads both times because 1/2 * 1/2 is 1/4.

with the cards, the chance of picking dog is 0.4. having two people pick one is like picking a card twice in a row. so 0.4 * 0.4 is 0.16, or 16%