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Annette [7]
2 years ago
8

Giải phương trình vi phân a. dy/dx b. y' + 2xy = 4x

Mathematics
1 answer:
WITCHER [35]2 years ago
8 0

Your first differential equation is incomplete, so I'll just look at the second one.

<em>y'</em> + 2<em>xy</em> = 4<em>x</em>

Multiply both sides by exp(<em>x</em> ²):

exp(<em>x</em> ²) <em>y'</em> + 2<em>x</em> exp(<em>x</em> ²) <em>y</em> = 4<em>x</em> exp(<em>x</em> ²)

The left side is the derivative of a product:

(exp(<em>x</em> ²) <em>y</em> )<em>'</em> = 4<em>x</em> exp(<em>x</em> ²)

Integrate both sides:

exp(<em>x</em> ²) <em>y</em> = ∫ 4<em>x</em> exp(<em>x</em> ²) d<em>x</em>

For the integral, substitute <em>u</em> = <em>x</em> ² and d<em>u</em> = 2<em>x</em> d<em>x</em> :

exp(<em>x</em> ²) <em>y</em> = 2 ∫ exp(<em>u</em>) d<em>u</em>

Solve for <em>y</em> :

exp(<em>x</em> ²) <em>y</em> = 2 exp(<em>u</em>) + <em>C</em>

exp(<em>x</em> ²) <em>y</em> = 2 exp(<em>x</em> ²) + <em>C</em>

<em>y</em> = 2 + <em>C</em> exp(-<em>x</em> ²)

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