Question

Find f(x) if y = f(x) satisfies
\frac(dy)(dx) = 45 yx^(14)
and the y -intercept of the curve y = f(x) is 3 .

Answer 1

i don't know squat about differential equations, but my first guess would be to divide both sides by \(y\) and integrate

Answer 2

\[\frac{y'}{y}=x^{14}\]
\[\ln(y)=\frac{x^{15}}{15}\] etc

Answer 3

oh i forgot the 45 sorry,
\[\ln(y)=3x^{15}\]

Answer 4

and so \(y=ce^{3x^{15}}\)

Answer 5

but i would get a second opinion, because this is a stab
looks good though, because you can check by differentiation

Answer 6

Yeah, nothing wrong with what ya did @satellite73

Answer 7

I agree, but would c in this case be equal to 3? because is the yintercept.

Answer 8

We don't need c to equal anything in this case. We don't have any initial condition to straight out find C.

Answer 9

ok, thanks!

Answer 10

Oh, there was an initial condition, my bad, haha.

Answer 11

Fail.

Answer 12

Nah, gotta get the full solution form first.

Answer 13

ok, you finish.

Answer 14

Might as well do the problem over, haha.

\[\frac{ dy }{ dx }=45yx ^{14}\]
\[\int\limits_{}^{}\frac{ dy }{ y }=45\int\limits_{}^{}x ^{14}\]
\[lny = (45)\frac{ x ^{15} }{ 15 }+C \]
\[e ^{lny}=e ^{3x ^{15}+C}\]
\[y=e ^{C}*e ^{3x ^{15}}\]
\[y=ce ^{3x ^{15}} \]
y-intercepts occur at x = 0. A y -intercept of 3 means we have the initial conditional of y(0) = 3
\[3=ce ^{3(0)^{15}}\rightarrow 3=c\]Particular Solution is then:
\[y=3e ^{3x ^{15}} \]

Answer 15

yeah that's what I thought

Answer 16

Thank you all!