Question

find particular solution y=f(x

Answer 1

its easy

Answer 2

\[\frac{ dy }{ dx }=6x^2-x^2y\]

Answer 3

\[\Large \frac{dy}{dx}=(6-y)x^2 \]
Or
\[ \Large \frac{1}{6-y}dy=x^2dx \] For \(y(x) \neq 6\)

Answer 4

\[\int\limits_{}^{}\frac{ 1 }{ 6-y }dy=x^2dx\]

Answer 5

Integrate both sides.

Answer 6

\[-lny=2x^3+c\]

Answer 7

@Spacelimbus

Answer 8

\[\Large - \ln |6-y(x)|=\frac{1}{3}x^3+C\prime  \]

Answer 9

they have given f(-1)=2

Answer 10

no calc

Answer 11

calculator

Answer 12

You know you have to solve it for y(x) if you can, of course - you could apply the initial conditions just now and solve for the constant. But I strongly recommend you to get the equation in explicit form if possible.

Answer 13

i got \[y(x)=ce ^{(x^3/x)}+6\]

Answer 14

Almost, try again. You should end up with:

\[\Large y(x)=6-Ce^{-\frac{1}{3}x^3} \]
As you can see, I said above that \(y(x) \neq 6\) which was important, otherwise I would have divided the differential equation through 0. This equation supports that statement, this equation only becomes 6 in the limit as x approaches infinity.