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Ordinary differential equation

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Differential equation containing derivatives with respect to only one variable

parabolic projectile motion showing velocity vector — Thetrajectory of aprojectile launched from acannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law.

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In mathematics, anordinary differential equation (ODE) is adifferential equation (DE) dependent on only a single independentvariable. As with any other DE, its unknown(s) consists of one (or more)function(s) and involves thederivatives of those functions.^[1] The term "ordinary" is used in contrast withpartial differential equations (PDEs) which may be with respect tomore than one independent variable,^[2] and, less commonly, in contrast withstochastic differential equations (SDEs) where the progression is random.^[3]

Theorem	Assumption	Conclusion
Peano existence theorem	$F {\displaystyle F}$ continuous	local existence only
Picard–Lindelöf theorem	$F {\displaystyle F}$ Lipschitz continuous	local existence and uniqueness

Differential equation	Solution method	General solution
First-order, separable in $x {\displaystyle x}$ and $y {\displaystyle y}$ (general case, see below for special cases)^[32] ${\begin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac {dy}{dx}}&=0\\P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy&=0\end{aligned}}$	Separation of variables (divide by $P_{2}Q_{1}$ ).	$\int ^{x}{\frac {P_{1}(\lambda )}{P_{2}(\lambda )}}\,d\lambda +\int ^{y}{\frac {Q_{2}(\lambda )}{Q_{1}(\lambda )}}\,d\lambda =C$
First-order, separable in $x {\displaystyle x}$ ^[30] ${\begin{aligned}{\frac {dy}{dx}}&=F(x)\\dy&=F(x)\,dx\end{aligned}}$	Direct integration.	$y=\int ^{x}F(\lambda )\,d\lambda +C$
First-order, autonomous, separable in $y {\displaystyle y}$ ^[30] ${\begin{aligned}{\frac {dy}{dx}}&=F(y)\\dy&=F(y)\,dx\end{aligned}}$	Separation of variables (divide by $F {\displaystyle F}$ ).	$x=\int ^{y}{\frac {d\lambda }{F(\lambda )}}+C$
First-order, separable in $x {\displaystyle x}$ and $y {\displaystyle y}$ ^[30] ${\begin{aligned}P(y){\frac {dy}{dx}}+Q(x)&=0\\P(y)\,dy+Q(x)\,dx&=0\end{aligned}}$	Integrate throughout.	$\int ^{y}P(\lambda )\,d\lambda +\int ^{x}Q(\lambda )\,d\lambda =C$

Differential equation	Solution method	General solution
First-order, homogeneous^[30] ${\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)$	Sety = ux, then solve by separation of variables inu andx.	$\ln(Cx)=\int ^{y/x}{\frac {d\lambda }{F(\lambda )-\lambda }}$
First-order, separable^[32] ${\begin{aligned}yM(xy)+xN(xy)\,{\frac {dy}{dx}}&=0\\yM(xy)\,dx+xN(xy)\,dy&=0\end{aligned}}$	Separation of variables (divide by $x y {\displaystyle xy}$ ).	$\ln(Cx)=\int ^{xy}{\frac {N(\lambda )\,d\lambda }{\lambda [N(\lambda )-M(\lambda )]}}$ If $N=M$ , the solution is $xy=C$ .
Exact differential, first-order^[30] ${\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}$ where ${\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}$	Integrate throughout.	${\begin{aligned}F(x,y)&=\int ^{x}M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\&=\int ^{y}N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}$ where $Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}M(\lambda ,y)\,d\lambda$ and $X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}N(x,\lambda )\,d\lambda$
Inexact differential, first-order^[30] ${\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}$ where ${\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}$	Integration factor $\mu (x,y)$ satisfying ${\frac {\partial (\mu M)}{\partial y}}={\frac {\partial (\mu N)}{\partial x}}$	If $\mu (x,y)$ can be found in a suitable way, then ${\begin{aligned}F(x,y)=&\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\=&\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}$ where $Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda$ and $X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda$

Differential equation	Solution method	General solution
First-order, linear, inhomogeneous, function coefficients^[30] ${\frac {dy}{dx}}+P(x)y=Q(x)$	Integrating factor: $e^{\int ^{x}P(\lambda )\,d\lambda }.$	$y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left[\int ^{x}e^{\int ^{\lambda }P(\varepsilon )\,d\varepsilon }Q(\lambda )\,d\lambda +C\right]$
Second-order, linear, inhomogeneous, function coefficients ${\frac {d^{2}y}{dx^{2}}}+2p(x){\frac {dy}{dx}}+\left(p(x)^{2}+p'(x)\right)y=q(x)$	Integrating factor: $e^{\int ^{x}P(\lambda )\,d\lambda }$	$y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left[\int ^{x}\left(\int ^{\xi }e^{\int ^{\lambda }P(\varepsilon )\,d\varepsilon }Q(\lambda )\,d\lambda \right)d\xi +C_{1}x+C_{2}\right]$
Second-order, linear, inhomogeneous, constant coefficients^[34] ${\frac {d^{2}y}{dx^{2}}}+b{\frac {dy}{dx}}+cy=r(x)$	Complementary function $y_{c}$ : assume $y_{c}(x)=e^{\alpha x}$ , substitute and solve polynomial in $\alpha$ , to find thelinearly independent functions $e^{\alpha _{j}x}$ . Particular integral $y_{p}$ : in general themethod of variation of parameters, though for very simple $r(x)$ inspection may work.^[30]	$y=y_{c}+y_{p}$ If $b^{2}>4c$ , then $y_{c}=C_{1}e^{-{\frac {x}{2}}\,\left(b+{\sqrt {b^{2}-4c}}\right)}+C_{2}e^{-{\frac {x}{2}}\,\left(b-{\sqrt {b^{2}-4c}}\right)}$ If $b^{2}=4c$ , then $y_{c}=(C_{1}x+C_{2})e^{-{\frac {bx}{2}}}$ If $b^{2}<4c$ , then $y_{c}=e^{-{\frac {bx}{2}}}\left[C_{1}\sin \left(x\,{\frac {\sqrt {4c-b^{2}}}{2}}\right)+C_{2}\cos \left(x\,{\frac {\sqrt {4c-b^{2}}}{2}}\right)\right]$
$n {\displaystyle n}$ th-order, linear, inhomogeneous, constant coefficients^[34] $\sum _{j=0}^{n}b_{j}{\frac {d^{j}y}{dx^{j}}}=r(x)$	Complementary function $y_{c}$ : assume $y_{c}(x)=e^{\alpha x}$ , substitute and solve polynomial in $\alpha$ , to find thelinearly independent functions $e^{\alpha _{j}x}$ . Particular integral $y_{p}$ : in general themethod of variation of parameters, though for very simple $r(x)$ inspection may work.^[30]	$y=y_{c}+y_{p}$ Since $\alpha _{j}$ are the solutions of thepolynomial ofdegree $n {\displaystyle n}$ : ${\textstyle \prod _{j=1}^{n}(\alpha -\alpha _{j})=0}$ , then:for $\alpha _{j}$ all different, $y_{c}=\sum _{j=1}^{n}C_{j}e^{\alpha _{j}x}$ for each root $\alpha _{j}$ repeated $k_{j}$ times, $y_{c}=\sum _{j=1}^{n}\left(\sum _{\ell =1}^{k_{j}}C_{j,\ell }x^{\ell -1}\right)e^{\alpha _{j}x}$ for some $\alpha _{j}$ complex, then setting $\alpha _{j}=\chi _{j}+i\gamma _{j}$ , and usingEuler's formula, allows some terms in the previous results to be written in the form $C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\varphi _{j})$ where $\varphi _{j}$ is an arbitrary constant (phase shift).

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Differential equations

Background

Definitions

General definition

System of ODEs

Solutions

Solutions of finite duration

Theories

Singular solutions

Reduction to quadratures

Fuchsian theory

Lie's theory

Sturm–Liouville theory

Existence and uniqueness of solutions

Local existence and uniqueness theorem simplified

Global uniqueness and maximum domain of solution

Reduction of order

Reduction to a first-order system

Summary of exact solutions

Separable equations

General first-order equations

General second-order equations

Linear to thenth order equations

Guessing solutions

Software for ODE solving

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