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rosijanka [135]
2 years ago
6

What is the particular solution to the differential equation dy/dx=(x+5)/y with the initial condition y(-4)=-6

Mathematics
1 answer:
Otrada [13]2 years ago
3 0

This equation is separable:

d<em>y</em>/d<em>x</em> = (<em>x</em> + 5) / <em>y</em>   →   <em>y</em> d<em>y</em> = (<em>x</em> + 5) d<em>x</em>

Integrate both sides to get

1/2 <em>y</em>² = 1/2 (<em>x</em> + 5)² + <em>C</em> … … … (if you substitute <em>u</em> = <em>x</em> + 5)

<u>or</u>

1/2 <em>y</em>² = 1/2 <em>x</em>² + 5<em>x</em> + <em>C</em> … … … (if you integrate term-by-term)

Given that <em>y</em> (-4) = -6, meaning <em>x</em> = -4 and <em>y</em> = -6, plug this in to solve for <em>C</em>. Note that whichever method you choose determines the particular value of <em>C</em>.

1/2 (-6)² = 1/2 (-4 + 5)² + <em>C</em>

18 = 1/2 + <em>C</em>

<em>C</em> = 35/2

→   1/2 <em>y</em>² = 1/2 (<em>x</em> + 5)² + 35/2

<u>or</u>

1/2 (-6)² = 1/2 (-4)² + 5(-4) + <em>C</em>

18 = 8 - 20 + <em>C</em>

<em>C</em> = 30

→   1/2 <em>y</em>² = 1/2 <em>x</em>² + 5<em>x</em> + 30

Both implicit solutions are correct and equivalent; the second one is just the expanded form of the first.

If you want an explicit solution, you can solve for <em>y</em>. But keep in mind that the initial condition tells you <em>y</em> < 0, so you should take the negative square root.

1/2 <em>y</em>² = 1/2 (<em>x</em> + 5)² + 35/2

<em>y</em>² = (<em>x</em> + 5)² + 35

<em>y</em> = -√((<em>x</em> + 5)² + 35)

<u>or</u>

<em>y</em> = -√(<em>x</em>² + 10<em>x</em> + 60)

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