# साधारण अवकल समीकरण

गणित में साधारण अवकल समीकरण (ordinary differential equation या ODE) उन अवकल समीकरणों को कहते हैं जिसमें केवल एक स्वतंत्र चर तथा उसके अवकलज मौजूद हों।

## उदाहरण

तोप से छोड़े गये गोले का पथ, साधारण अवकल समीकरण के रूप में वर्णित न्यूटन के द्वितीय नियम द्वारा निरूपित वक्र ही है।

साधारण अवकल समीकरण का एक उदाहरण नियत द्रव्यमान m वाली किसी वस्तु की गति का वर्णन करने वाला न्यूटन का गति का दूसरा नियम है-

$F(x(t))\ =m{\frac {d^{2}x(t)}{dt^{2}}}$

जिसमें बल F वस्तु की स्थिति x (t) तथा समय t पर निर्भर है। इसमें अज्ञात फलन x (t) समीकरण के दोनों तरफ विद्यमान है।

अन्य उदाहरण:

• ${\begin{cases}y'=y\\y(0)=2\end{cases}}$

जिसका हल है : $y=2e^{x}\,\!$ .

• $y'=3y^{2}-2x^{3}+4\,$
• ${\frac {y}{1+x}}+y'+(1+x)y^{4}+3y''=0$
• $\left({\frac {dy}{dx}}\right)^{2}-8x{\frac {dy}{dx}}+5xy^{3}-x=0$ , जिसको इस तरह भी लिख सकते हैं: $(y')^{2}+8xy'+5xy^{3}-x=0\,$
• $8d^{2}y=dx\,$

## यथातथ्य (exact) हल

कुछ अवकल समीकरणों का हल ठीक-ठीक तथा बन्द स्वरूप (exact and closed form) में लिखा जा सकता है। यहाँ कुछ प्रकार के महत्वपूर्ण अवकल समीकरण दिये गये हैं-

अवकल समीकरण हल की विधि सामान्य हल
चर अलग करने योग्य समीकरण (Separable equations)
First-order, separable in x and y (general case, see below for special cases)

$P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac {dy}{dx}}=0\,\!$

$P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy=0\,\!$

Separation of variables (divide by P2Q1). $\int ^{x}{\frac {P_{1}(\lambda )}{P_{2}(\lambda )}}\,d\lambda +\int ^{y}{\frac {Q_{2}(\lambda )}{Q_{1}(\lambda )}}\,d\lambda =C\,\!$
First-order, separable in x

${\frac {dy}{dx}}=F(x)\,\!$

$dy=F(x)\,dx\,\!$

Direct integration. $y=\int ^{x}F(\lambda )\,d\lambda +C\,\!$
First-order, autonomous, separable in y

${\frac {dy}{dx}}=F(y)\,\!$

$dy=F(y)\,dx\,\!$

Separation of variables (divide by F). $x=\int ^{y}{\frac {d\lambda }{F(\lambda )}}+C\,\!$
First-order, separable in x and y

$P(y){\frac {dy}{dx}}+Q(x)=0\,\!$

$P(y)\,dy+Q(x)\,dx=0\,\!$

Integrate throughout. $\int ^{y}P(\lambda )\,{d\lambda }+\int ^{x}Q(\lambda )\,d\lambda =C\,\!$
General first-order equations
First-order, homogeneous

${\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)\,\!$

Set y = ux, then solve by separation of variables in u and x. $\ln(Cx)=\int ^{y/x}{\frac {d\lambda }{F(\lambda )-\lambda }}\,\!$
First-order, separable

$yM(xy)+xN(xy)\,{\frac {dy}{dx}}=0\,\!$

$yM(xy)\,dx+xN(xy)\,dy=0\,\!$

Separation of variables (divide by 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

$M(x,y){\frac {dy}{dx}}+N(x,y)=0\,\!$

$M(x,y)\,dy+N(x,y)\,dx=0\,\!$

where ${\frac {\partial M}{\partial x}}={\frac {\partial N}{\partial y}}\,\!$

Integrate throughout. {\begin{aligned}F(x,y)&=\int ^{y}M(x,\lambda )\,d\lambda +\int ^{x}N(\lambda ,y)\,d\lambda \\&+Y(y)+X(x)=C\end{aligned}}\,\!

where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.

Inexact differential, first-order

$M(x,y){\frac {dy}{dx}}+N(x,y)=0\,\!$

$M(x,y)\,dy+N(x,y)\,dx=0\,\!$

where ${\frac {\partial M}{\partial x}}\neq {\frac {\partial N}{\partial y}}\,\!$

Integration factor μ(x, y) satisfying

${\frac {\partial (\mu M)}{\partial x}}={\frac {\partial (\mu N)}{\partial y}}\,\!$

If μ(x, y) can be found:

{\begin{aligned}F(x,y)&=\int ^{y}\mu (x,\lambda )M(x,\lambda )\,d\lambda +\int ^{x}\mu (\lambda ,y)N(\lambda ,y)\,d\lambda \\&+Y(y)+X(x)=C\\\end{aligned}}\,\!

General second-order equations
Second-order, autonomous

${\frac {d^{2}y}{dx^{2}}}=F(y)\,\!$

Multiply equation by 2dy/dx, substitute $2{\frac {dy}{dx}}{\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)^{2}\,\!$ , then integrate twice. $x=\pm \int ^{y}{\frac {d\lambda }{\sqrt {2\int ^{\lambda }F(\epsilon )\,d\epsilon +C_{1}}}}+C_{2}\,\!$
Linear equations (up to nth order)
First-order, linear, inhomogeneous, function coefficients

${\frac {dy}{dx}}+P(x)y=Q(x)\,\!$

Integrating factor: $e^{\int ^{x}P(\lambda )\,d\lambda }$ . $y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left[\int ^{x}e^{\int ^{\lambda }P(\epsilon )\,d\epsilon }Q(\lambda )\,{d\lambda }+C\right]$
Second-order, linear, inhomogeneous, constant coefficients

${\frac {d^{2}y}{dx^{2}}}+b{\frac {dy}{dx}}+cy=r(x)\,\!$

Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions $e^{\alpha _{j}x}$ .

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.

$y=y_{c}+y_{p}$

If b2 > 4c, then:

$y_{c}=C_{1}e^{\left(-b+{\sqrt {b^{2}-4c}}\right){\frac {x}{2}}}+C_{2}e^{-\left(b+{\sqrt {b^{2}-4c}}\right){\frac {x}{2}}}\,\!$

If b2 = 4c, then:

$y_{c}=(C_{1}x+C_{2})e^{-bx/2}\,\!$

If b2 < 4c, then:

$y_{c}=e^{-b{\frac {x}{2}}}\left[C_{1}\sin {\left({\sqrt {\left|b^{2}-4c\right|}}{\frac {x}{2}}\right)}+C_{2}\cos {\left({\sqrt {\left|b^{2}-4c\right|}}{\frac {x}{2}}\right)}\right]\,\!$

nth-order, linear, inhomogeneous, constant coefficients

$\sum _{j=0}^{n}b_{j}{\frac {d^{j}y}{dx^{j}}}=r(x)\,\!$

Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions $e^{\alpha _{j}x}$ .

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.

$y=y_{c}+y_{p}$

Since αj are the solutions of the polynomial of degree n: $\prod _{j=1}^{n}\left(\alpha -\alpha _{j}\right)=0\,\!$ , then:

for αj all different,

$y_{c}=\sum _{j=1}^{n}C_{j}e^{\alpha _{j}x}\,\!$

for each root αj repeated kj times,

$y_{c}=\sum _{j=1}^{n}\left(\sum _{\ell =1}^{k_{j}}C_{\ell }x^{\ell -1}\right)e^{\alpha _{j}x}\,\!$

for some αj complex, then setting α = χj + iγj, and using Euler's formula, allows some terms in the previous results to be written in the form

$C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\phi _{j})\,\!$

where ϕj is an arbitrary constant (phase shift).

## सन्दर्भ

1. Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7
2. सन्दर्भ त्रुटि: <ref> का गलत प्रयोग; EDEBVP नाम के संदर्भ में जानकारी नहीं है।
3. Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, ISBN 0-7135-1594-5
4. Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3