First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy
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Theorem
The first order ordinary differential equation:
- $(1): \quad \paren {2 x y^3 + y \cos x} \d x + \paren {3 x^2 y^2 + \sin x} \d y = 0$
is an exact differential equation with solution:
- $x^2 y^3 + y \sin x = C$
This can also be presented as:
- $\dfrac {\d y} {\d x} = -\dfrac {2 x y^3 + y \cos x} {3 x^2 y^2 + \sin x}$
Proof
Let:
- $\map M {x, y} = 2 x y^3 + y \cos x$
- $\map N {x, y} = 3 x^2 y^2 + \sin x$
Then:
\(\ds \frac {\partial M} {\partial y}\) | \(=\) | \(\ds 2 x \cdot 3 y^2 + \cos x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 x y^2 + \cos x\) | ||||||||||||
\(\ds \frac {\partial N} {\partial x}\) | \(=\) | \(\ds 3 x^2 \cdot 2 y^2 + \cos x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 x y^2 + \cos x\) |
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ and $(1)$ is seen by definition to be exact.
By Solution to Exact Differential Equation, the solution to $(1)$ is:
- $\map f {x, y} = C$
where:
\(\ds \dfrac {\partial f} {\partial x}\) | \(=\) | \(\ds \map M {x, y}\) | ||||||||||||
\(\ds \dfrac {\partial f} {\partial y}\) | \(=\) | \(\ds \map N {x, y}\) |
Hence:
\(\ds f\) | \(=\) | \(\ds \map M {x, y} \rd x + \map g y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {2 x y^3 + y \cos x} \rd x + \map g y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^2 y^3 + y \sin x + \map g y\) |
and:
\(\ds f\) | \(=\) | \(\ds \int \map N {x, y} \rd y + \map h x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {3 x^2 y^2 + \sin x} \rd y + \map h x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^2 y^3 + y \sin x + \map h x\) |
Thus:
- $\map f {x, y} = x^2 y^3 + y \sin x$
and by Solution to Exact Differential Equation, the solution to $(1)$ is:
- $x^2 y^3 + y \sin x = C$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.8$: Exact Equations: Problem $10$