§22.13 Derivatives and Differential Equations

Permalink:: http://dlmf.nist.gov/22.13

§22.13(i) Derivatives
§22.13(ii) First-Order Differential Equations
§22.13(iii) Second-Order Differential Equations

§22.13(i) Derivatives

Notes:: See Lawden (1989, §2.5), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 491–494).
Keywords:: Jacobian elliptic functions, derivatives
Permalink:: http://dlmf.nist.gov/22.13.i

Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

$\frac{d}{dz}(\mathop{\mathrm{sn}\/}\nolimits z)$ $\;=\;$	$\mathop{\mathrm{cn}\/}\nolimits z\mathop{\mathrm{dn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{dc}\/}\nolimits z)$ =	${k^{{\prime}}}^{2}\mathop{\mathrm{sc}\/}\nolimits z\mathop{\mathrm{nc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{cn}\/}\nolimits z)$ $\;=\;$	$-\mathop{\mathrm{sn}\/}\nolimits z\mathop{\mathrm{dn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{nc}\/}\nolimits z)$ =	$\mathop{\mathrm{sc}\/}\nolimits z\mathop{\mathrm{dc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{dn}\/}\nolimits z)$ $\;=\;$	$-k^{2}\mathop{\mathrm{sn}\/}\nolimits z\mathop{\mathrm{cn}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{sc}\/}\nolimits z)$ =	$\mathop{\mathrm{dc}\/}\nolimits z\mathop{\mathrm{nc}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{cd}\/}\nolimits z)$ $\;=\;$	$-{k^{{\prime}}}^{2}\mathop{\mathrm{sd}\/}\nolimits z\mathop{\mathrm{nd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{ns}\/}\nolimits z)$ =	$-\mathop{\mathrm{ds}\/}\nolimits z\mathop{\mathrm{cs}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{sd}\/}\nolimits z)$ $\;=\;$	$\mathop{\mathrm{cd}\/}\nolimits z\mathop{\mathrm{nd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{ds}\/}\nolimits z)$ =	$-\mathop{\mathrm{cs}\/}\nolimits z\mathop{\mathrm{ns}\/}\nolimits z$
$\frac{d}{dz}(\mathop{\mathrm{nd}\/}\nolimits z)$ $\;=\;$	$k^{2}\mathop{\mathrm{sd}\/}\nolimits z\mathop{\mathrm{cd}\/}\nolimits z$	$\frac{d}{dz}(\mathop{\mathrm{cs}\/}\nolimits z)$ =	$-\mathop{\mathrm{ns}\/}\nolimits z\mathop{\mathrm{ds}\/}\nolimits z$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: Table 16.16
Referenced by:: §22.13(i)
Permalink:: http://dlmf.nist.gov/22.13.T1

Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. (The modulus $k$ is suppressed throughout the table.)

For alternative, and symmetric, formulations of these results see Carlson (2004, 2006a).

§22.13(ii) First-Order Differential Equations

Notes:: See Whittaker and Watson (1927, pp. 491–498).
Keywords:: Jacobian elliptic functions, differential equations
Permalink:: http://dlmf.nist.gov/22.13.ii

22.13.1 $\left(\frac{d}{dz}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)\right)^{2}=\left(1-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Referenced by:: §22.13(iii), §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.13.E1
Encodings:: TeX, pMML, png

22.13.2 $\left(\frac{d}{dz}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)\right)^{2}={\left(1-{\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(z,k\right)\right)}{\left({k^{{\prime}}}^{2}+k^{2}{\mathop{\mathrm{cn}\/}\nolimits^{{2}}}\left(z,k\right)\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E2
Encodings:: TeX, pMML, png

22.13.3 $\left(\frac{d}{dz}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)\right)^{2}=\left(1-{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left({\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z,k\right)-{k^{{\prime}}}^{2}\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E3
Encodings:: TeX, pMML, png

22.13.4 $\left(\frac{d}{dz}\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)\right)^{2}=\left(1-{\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1-k^{2}{\mathop{\mathrm{cd}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E4
Encodings:: TeX, pMML, png

22.13.5 $\left(\frac{d}{dz}\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)\right)^{2}={\left(1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(z,k\right)\right)}{\left(1+k^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}\left(z,k\right)\right)},$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E5
Encodings:: TeX, pMML, png

22.13.6 $\left(\frac{d}{dz}\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)\right)^{2}=\left({\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z,k\right)-1\right)\left(1-{k^{{\prime}}}^{2}{\mathop{\mathrm{nd}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E6
Encodings:: TeX, pMML, png

22.13.7 $\left(\frac{d}{dz}\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)\right)^{2}=\left({\mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)-1\right)\left({\mathop{\mathrm{dc}\/}\nolimits^{{2}}}\left(z,k\right)-k^{2}\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E7
Encodings:: TeX, pMML, png

22.13.8 $\left(\frac{d}{dz}\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)\right)^{2}={\left(k^{2}+{k^{{\prime}}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z,k\right)\right)}{\left({\mathop{\mathrm{nc}\/}\nolimits^{{2}}}\left(z,k\right)-1\right)},$

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E8
Encodings:: TeX, pMML, png

22.13.9 $\left(\frac{d}{dz}\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)\right)^{2}=\left(1+{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left(1+{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E9
Encodings:: TeX, pMML, png

22.13.10 $\left(\frac{d}{dz}\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)\right)^{2}=\left({\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(z,k\right)-k^{2}\right)\left({\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(z,k\right)-1\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E10
Encodings:: TeX, pMML, png

22.13.11 $\left(\frac{d}{dz}\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)\right)^{2}=\left({\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(z,k\right)-{k^{{\prime}}}^{2}\right)\left(k^{2}+{\mathop{\mathrm{ds}\/}\nolimits^{{2}}}\left(z,k\right)\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E11
Encodings:: TeX, pMML, png

22.13.12 $\left(\frac{d}{dz}\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)\right)^{2}=\left(1+{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(z,k\right)\right)\left({k^{{\prime}}}^{2}+{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(z,k\right)\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E12
Encodings:: TeX, pMML, png

For alternative, and symmetric, formulations of these results see Carlson (2006a).

§22.13(iii) Second-Order Differential Equations

Notes:: Equations (22.13.13)–(22.13.24) are obtained by differentiation of (22.13.1)–(22.13.12).
Keywords:: Jacobian elliptic functions, differential equations
Permalink:: http://dlmf.nist.gov/22.13.iii

22.13.13 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=-(1+k^{2})\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)+2k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Referenced by:: §22.10(i), §22.10(ii), §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E13
Encodings:: TeX, pMML, png

22.13.14 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=-({k^{{\prime}}}^{2}-k^{2})\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)-2k^{2}{\mathop{\mathrm{cn}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.10(i), §22.10(ii)
Permalink:: http://dlmf.nist.gov/22.13.E14
Encodings:: TeX, pMML, png

22.13.15 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=(1+{k^{{\prime}}}^{2})\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)-2{\mathop{\mathrm{dn}\/}\nolimits^{{3}}}\left(z,k\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.10(i), §22.10(ii)
Permalink:: http://dlmf.nist.gov/22.13.E15
Encodings:: TeX, pMML, png

22.13.16 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)=-(1+k^{2})\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)+2k^{2}{\mathop{\mathrm{cd}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E16
Encodings:: TeX, pMML, png

22.13.17 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)=(k^{2}-{k^{{\prime}}}^{2})\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)-2k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E17
Encodings:: TeX, pMML, png

22.13.18 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)=(1+{k^{{\prime}}}^{2})\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)-2{k^{{\prime}}}^{2}{\mathop{\mathrm{nd}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E18
Encodings:: TeX, pMML, png

22.13.19 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)=-(1+k^{2})\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{dc}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E19
Encodings:: TeX, pMML, png

22.13.20 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)=(k^{2}-{k^{{\prime}}}^{2})\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)+2{k^{{\prime}}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E20
Encodings:: TeX, pMML, png

22.13.21 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)=(1+{k^{{\prime}}}^{2})\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)+2{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E21
Encodings:: TeX, pMML, png

22.13.22 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)=-(1+k^{2})\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{ns}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E22
Encodings:: TeX, pMML, png

22.13.23 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)=(k^{2}-{k^{{\prime}}}^{2})\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{ds}\/}\nolimits^{{3}}}\left(z,k\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E23
Encodings:: TeX, pMML, png

22.13.24 $\frac{{d}^{2}}{{dz}^{2}}\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)=(1+{k^{{\prime}}}^{2})\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)+2{\mathop{\mathrm{cs}\/}\nolimits^{{3}}}\left(z,k\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.13(iii)
Permalink:: http://dlmf.nist.gov/22.13.E24
Encodings:: TeX, pMML, png

For alternative, and symmetric, formulations of these results see Carlson (2006a).

§22.13 Derivatives and Differential Equations

Contents

§22.13(i) Derivatives

§22.13(ii) First-Order Differential Equations

§22.13(iii) Second-Order Differential Equations