Fungrim entry: f1691f

y''(z) + {a}^{2} y(z) + b = 0\; \text{ where } y(z) = {c}_{1} \sin\!\left(a z\right) + {c}_{2} \cos\!\left(a z\right) - \frac{b}{{a}^{2}}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}

TeX:

y''(z) + {a}^{2} y(z) + b = 0\; \text{ where } y(z) = {c}_{1} \sin\!\left(a z\right) + {c}_{2} \cos\!\left(a z\right) - \frac{b}{{a}^{2}}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f1691f"),
    Formula(Where(Equal(Add(Add(ComplexDerivative(y(z), For(z, z, 2)), Mul(Pow(a, 2), y(z))), b), 0), Equal(y(z), Sub(Add(Mul(Subscript(c, 1), Sin(Mul(a, z))), Mul(Subscript(c, 2), Cos(Mul(a, z)))), Div(b, Pow(a, 2)))))),
    Variables(z, a, b, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(z, CC), Element(a, SetMinus(CC, Set(0))), Element(b, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC))))

Topics using this entry

Sine