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Strike441 [17]
2 years ago
14

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx = f(Ax + By + C), which

is given in (5) of Section 2.5. dy/dx = 4 + (y − 4x + 6)^1/2
Mathematics
1 answer:
Mariana [72]2 years ago
7 0

d<em>y</em>/d<em>x</em> = 4 + √(<em>y</em> - 4<em>x</em> + 6)

Make a substitution of <em>v(x)</em> = <em>y(x)</em> - 4<em>x</em> + 6, so that d<em>v</em>/d<em>x</em> = d<em>y</em>/d<em>x</em> - 4. Then the DE becomes

d<em>v</em>/d<em>x</em> + 4 = 4 + √<em>v</em>

d<em>v</em>/d<em>x</em> = √<em>v</em>

which is separable as

d<em>v</em>/√<em>v</em> = d<em>x</em>

Integrating both sides gives

2√<em>v</em> = <em>x</em> + <em>C</em>

Get the solution back in terms of <em>y</em> :

2√(<em>y</em> - 4<em>x</em> + 6) = <em>x</em> + <em>C</em>

You can go on to solve for <em>y</em> explicitly if you want.

√(<em>y</em> - 4<em>x</em> + 6) = <em>x</em>/2 + <em>C</em>

<em>y</em> - 4<em>x</em> + 6 = (<em>x</em>/2 + <em>C </em>)²

<em>y</em> = 4<em>x</em> - 6 + (<em>x</em>/2 + <em>C </em>)²

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