First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y
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Theorem
The first order ordinary differential equation:
- $(1): \quad x^2 \dfrac {\d y} {\d x} = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$
is a homogeneous differential equation with solution:
- $y = x \tan C x^3$
Proof
Let:
- $\map M {x, y} = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$
- $\map N {x, y} = x^2$
Put $t x, t y$ for $x, y$:
\(\ds \map M {t x, t y}\) | \(=\) | \(\ds 3 \paren {\paren {t x}^2 + \paren {t y}^2} \arctan \dfrac {t y} {t x} + t x t y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds t^2 \paren {3 \paren {x^2 + y^2} \arctan \dfrac y x + x y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds t^2 \, \map M {x, y}\) |
\(\ds \map N {t x, t y}\) | \(=\) | \(\ds \paren {t x}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds t^2 \paren {x^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds t^2 \, \map N {x, y}\) |
Thus both $M$ and $N$ are homogeneous functions of degree $2$.
Thus, by definition, $(1)$ is a homogeneous differential equation.
By Solution to Homogeneous Differential Equation, its solution is:
- $\ds \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$
where:
- $\map f {x, y} = \dfrac {3 \paren {x^2 + y^2} \arctan \dfrac y x + x y} {x^2}$
Thus:
\(\ds \ln x\) | \(=\) | \(\ds \int \frac {\d z} {\dfrac {3 \paren {1^2 + z^2} \arctan \frac z 1 + z} {1^2} - z} + C_1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d z} {3 \paren {1 + z^2} \arctan z} + C_1\) |
Substituting $u = \arctan z$:
\(\ds \dfrac {\d u} {\d x}\) | \(=\) | \(\ds \frac 1 {z^2 + 1}\) | Derivative of Arctangent Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d z} {\d u}\) | \(=\) | \(\ds z^2 + 1\) |
Hence:
\(\ds \ln x\) | \(=\) | \(\ds \int \frac {\d z} {3 \paren {1 + z^2} \arctan z} + C_1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d u} {3 u} + C_1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 3 \ln u + C_1\) | Primitive of Reciprocal | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln x^3 + \ln C\) | \(=\) | \(\ds \ln u\) | putting $\ln C = - C_1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds C x^3\) | \(=\) | \(\ds u\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds C x^3\) | \(=\) | \(\ds \arctan \frac y x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds x \tan C x^3\) |
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.7$: Homogeneous Equations: Problem $1 \ \text{(c)}$