anonymous
  • anonymous
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 .
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
i don't know squat about differential equations, but my first guess would be to divide both sides by \(y\) and integrate
anonymous
  • anonymous
\[\frac{y'}{y}=x^{14}\] \[\ln(y)=\frac{x^{15}}{15}\] etc
anonymous
  • anonymous
oh i forgot the 45 sorry, \[\ln(y)=3x^{15}\]

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anonymous
  • anonymous
and so \(y=ce^{3x^{15}}\)
anonymous
  • anonymous
but i would get a second opinion, because this is a stab looks good though, because you can check by differentiation
Psymon
  • Psymon
Yeah, nothing wrong with what ya did @satellite73
anonymous
  • anonymous
I agree, but would c in this case be equal to 3? because is the yintercept.
Psymon
  • Psymon
We don't need c to equal anything in this case. We don't have any initial condition to straight out find C.
anonymous
  • anonymous
ok, thanks!
Psymon
  • Psymon
Oh, there was an initial condition, my bad, haha.
Psymon
  • Psymon
Fail.
Psymon
  • Psymon
Nah, gotta get the full solution form first.
Loser66
  • Loser66
ok, you finish.
Psymon
  • Psymon
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: \[y=3e ^{3x ^{15}} \]
anonymous
  • anonymous
yeah that's what I thought
anonymous
  • anonymous
Thank you all!

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