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irga5000 [103]
3 years ago
8

given y>0 and dy/dx=(3x^2+4x)/yif the point (1,√10) is on the graph relating x and y, then what is y when x=0

Mathematics
1 answer:
jonny [76]3 years ago
4 0
\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{3x^2+4x}y\iff y\,\mathrm dy=(3x^2+4x)\,\mathrm dx
\implies\displaystyle\int y\,\mathrm dy=\int(3x^2+4x)\,\mathrm dx
\implies \dfrac12y^2=x^3+2x^2+C

When x=1 you have y=\sqrt{10}, so

\dfrac12(\sqrt{10})^2=1^3+2(1)^2+C\implies 5=1+2+C\implies C=2

and so the particular solution to the ODE is

\dfrac12y^2=x^3+2x^2+2

Then when x=0, you get

\dfrac12y^2=0^3+2(0)^2+2=2
\implies y^2=4
\implies y=2

where we omitted the negative root because it's given that y>0.
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