{tocify} $title={Table of Contents}
Limits & Derivatives
Algebra of Limits
For a function\(f(x)\) limits exists if and only
\[\lim_{{x \to a^{+}}}f(x)=\lim_{{x \to a^{-}}}f(x);\]
\[\lim_{{x \to a^{+}}}f(x)=\lim_{{h \to 0}}f(a+h)\]
\[\lim_{{x \to a^{-}}}f(x)=\lim_{{h \to 0}}f(a-h)\]
\[\lim_{{x \to a}}f(x)=\lim_{{h \to 0}}f(a+h)\]
\[\lim_{{x\to a}}(f(x)\pm g(x))=\lim_{{x\to a}}f(x)\pm\lim_{{x\to a}}g(x)\]
\[\lim_{{x\to a}}(f(x).g(x))=\lim_{{x\to a}}f(x).\lim_{{x\to a}}g(x)\]
\[\lim_{{x\to a}}\frac{f(x)}{g(x)}=\frac{\lim_{{x\to a}}f(x)}{\lim_{{x\to a}}g(x)};\lim_{{x\to a}}g(x)\neq0\]
\[\lim_{{x\to a}}f(x)^{g(x)}=\lim_{{x\to a}}f(x)^{\lim_{{x\to a}}g(x)}\]
Basic Limits
\[\lim_{{x\to o}}\frac{x^{n}-a^{n}}{x-a}=na^{n-1}\]
\[\lim_{{x\to o}}\frac{x^{n}-a^{n}}{x^{m}-a^{m}}=\frac{n}{m}a^{n-m}\]
\[\lim_{{x\to o}}\frac{sinx}{x}=1\]
\[\lim_{{x\to o}}\begin{bmatrix}\frac{sinx}{x}\end{bmatrix}=0\]
\[where \begin{bmatrix}x\end{bmatrix} denotes\text{ }GIF\]
\[\lim_{{x\to o}}\frac{tanx}{x}=1\]
\[\lim_{{x\to o}}\begin{bmatrix}\frac{tanx}{x}\end{bmatrix}=1\]
\[where\begin{bmatrix}x\end{bmatrix} denotes\text{ }GIF\]
\[\lim_{{x\to o}}\frac{sin^{-1}x}{x}=1\]
\[\lim_{{x\to o}}\frac{tan^{-1}x}{x}=1\]
\[\lim_{{x\to o}}\frac{log_{e}(1+x)}{x}=1\]
\[\lim_{{x\to o}}\frac{log_{a}(1+x)}{x}=\frac{1}{log_{e}a}\]
\[\lim_{{x\to o}}\frac{e^{x}-1}{x}=1\]
\[\lim_{{x\to o}}\frac{a^{x}-1}{x}=log_{e}a\]
\[\lim_{{f(x)\to 0}}(1+f(x))^{\frac{1}{f(x)}}=e^{1};1^{\infty}form\]
\[\lim_{{f(x)\to 0}}(f(x))^{\frac{1}{f(x)}}=e^{-1}=\frac{1}{e};0^{\infty}form\]
\[\lim_{{f(x)\to 0}}(f(x))^{f(x)}=1;0^{0}\]
\[\lim_{{f(x)\to 0}}(\frac{1}{f(x)})^{f(x)}=1;{\infty}^{0}\]
I'Hopital's Rule
Applicable for indeterminate forms \(\frac{0}{0}\frac{\infty}{\infty}\)
\[\lim_{{x\to a}}\frac{f(x)}{g(x)}=\lim_{{x\to a}}\frac{f^{|}(x)}{g^{|}(x)}=\lim_{{x\to a}}\frac{f^{||}(x)}{g^{||}(x)}=...\]
Expansion Series
\[sinx=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}..\]
\[sinx=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}\]
\[cosx=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}..\]
\[cosx=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}\]
\[e^{x}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}..\]
\[e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\]
\[log_{e}(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}..\]
\[log_{e}(1+x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{n+1}}{(n+1)}\]
\[(1+x)^{n}=1+\frac{nx}{1!}+\frac{n(n-1)x^{2}}{2!}+\frac{n(n-1)(n-2)x^{3}}{3!}...\]
Taylor Series:
\[f(x)=f(a)+\frac{f^{|}(a)}{1!}(x-a)+\frac{f^{||}(a)}{2!}(x-a)^{2}...\]
\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}\]
Maclaurin Series:
\[f(x)=f(0)+\frac{f^{|}(0)}{1!}(x)+\frac{f^{||}(a)}{2!}(x)^{2}...\]
\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}\]
Continuity & Differentiability
For a function \(f(x)\) to be continous at \(a\) the following condition need to be satisfied
\[\lim_{{x\to a^{+}}}f(x)=\lim_{{x\to a^{-}}}f(x)=f(a)\]
For a function \(f(x)\) to be continous in the interval \[\begin{bmatrix}a,b\end{bmatrix}\]
\[\lim_{{x\to a^{+}}}f(x)=f(a)\]
\[\lim_{{x\to b^{-}}}f(x)=f(b)\]
For a function \(f(x)\) to be differentiable at \(a\) the following condition need to be satisfied
\[\lim_{{h\to 0^{+}}}\frac{f(x+h)-f(x)}{h}=f^{|}(x)\]
\[\lim_{{h\to 0^{-}}}\frac{f(x+h)-f(x)}{h}=f^{|}(x)\]
Methods of Differentiation
Algebra of Differentiation
\[\frac{d}{dx}(u\pm v)=u^{|}\pm v^{|}\]
\[\frac{d}{dx}(uv)=vu^{|}+uv^{|}\]
\[\frac{d}{dx}(\frac{u}{v})=\frac{vu^{|}-uv^{|}}{v^{2}}\]
\[\frac{d}{dx}(uvw)=u^{|}vw+uv^{|}w+uvw^{|}\]
\[where\text{ }\frac{du}{dx}=u^{|};\frac{dv}{dx}=v^{|};\frac{dw}{dx}=w^{|}\]
Basic Formulae
\[\frac{d}{dx}(constant)=0\]
\[\frac{d}{dx}(x^{n})=nx^{n-1}\]
\[\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}\]
\[\frac{d}{dx}(log_{e}x)=\frac{1}{x}\]
\[\frac{d}{dx}(log_{a}x)=\frac{1}{xlog_{e}a}\]
\[\frac{d}{dx}(e^{x})=e^{x}\]
\[\frac{d}{dx}(a^{x})=a^{x}log_{e}a\]
\[\frac{d}{dx}(sinx)=cosx\]
\[\frac{d}{dx}(cosx)=-sinx\]
\[\frac{d}{dx}(tanx)=sec^{2}x=\frac{1}{cos^{2}x}\]
\[\frac{d}{dx}(cotx)=-cosec^{2}x=-\frac{1}{sin^{2}x}\]
\[\frac{d}{dx}(secx)=secxtanx\]
\[\frac{d}{dx}(cosecx)=-cosecxcotx\]
\[\frac{d}{dx}(sin^{-1}x)=\frac{1}{\sqrt{1-x^{2}}}=-\frac{d}{dx}(cos^{-1}x)\]
\[\frac{d}{dx}(cos^{-1}x)=-\frac{1}{\sqrt{1-x^{2}}}=-\frac{d}{dx}(sin^{-1}x)\]
\[\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^{2}}=-\frac{d}{dx}(cot^{-1}x)\]
\[\frac{d}{dx}(cot^{-1}x)=-\frac{1}{1+x^{2}}=-\frac{d}{dx}(tan^{-1}x)\]
\[\frac{d}{dx}(sec^{-1}x)=\frac{1}{x\sqrt{x^{2}-1}}=-\frac{d}{dx}(cosec^{-1}x)\]
\[\frac{d}{dx}(cosec^{-1}x)=-\frac{1}{x\sqrt{x^{2}-1}}=-\frac{d}{dx}(sec^{-1}x)\]
\[\frac{d}{dx}(sin^{2}x)=2sinxcosx=sin2x\]
\[\frac{d}{dx}(cos^{2}x)=-2sinxcosx=-sin2x\]
\[\frac{d}{dx}(f(x))^{g(x)}=f(x)^{g(x)}\begin{bmatrix}\frac{g(x)f^{|}(x)}{f(x)}+[\text{ }log_{e}f(x)\text{ }]g^{|}(x)\end{bmatrix}\]
Applications of Derivatives
Mensuration formulae
\[V_{cuboid}=l\times b\times h\]
\[V_{prism}=A_{base}\times h\]
\[V_{pyramid}=\frac{1}{3}A_{base}\times h\]
\[V_{cone}=\frac{1}{3}\pi r^{2}\times h\]
\[V_{sphere}=\frac{4}{3}\pi r^{3}\]
\[V_{hemisphere}=\frac{2}{3}\pi r^{3}\]
\[V_{frustum}=\frac{1}{3}\pi h(r^{2}+R^{2}+rR)\]
\[TSA_{sphere}={4}\pi r^{2}\]
\[TSA_{cone}=\pi r(r+\sqrt{r^{2}+h^{2}})\]
\[TSA_{cone}=\pi r\sqrt{r^{2}+h^{2}}\]
\[TSA_{cuboid}=2(lb\times bh\times hl)\]
\[LSA_{cuboid}=2(l+b)\times h\]
\[LSA_{prism}=P_{base}\times h\]
\[TSA_{prism}=P_{base}\times h+2A_{base}\]
\[LSA_{pyramid}=\frac{1}{2}P_{base}\times h_{slant}\]
\[TSA_{pyramid}=\frac{1}{2}P_{base}\times h_{slant}+\frac{P^{2}_{base}}{16}\]
\[LSA_{cylinder}=2\pi r\times h\]
\[LSA_{frustum}=\pi(\sqrt{h^{2}+(R-r)^{2}})(R+r)\]
\[TSA_{frustum}=CSA_{frustum}+\pi(R^{2}+r^{2})\]
\[A_{sector}=\frac{1}{2}r^{2}\theta\text{ }where\text{ }\theta\text{ in radians }\]
Indefinite Integration
Basic formulas
\[\int x^{n}dx=\frac{x^{n+1}}{n+1}+C\]
\[\int \frac{1}{\sqrt{x}}dx=2\sqrt{x}+C\]
\[\int sin x dx =-cos x + C\]
\[\int cos x dx =sin x + C\]
\[\int cosecx dx =ln |cosecx-cotx| + C\]
\[\int secx dx =ln |secx+tanx| + C\]
\[\int tanx dx =-ln |cosx| + C\]
\[\int cotx dx =ln |sinx| + C\]
\[\int cosec^{2}x dx =-cotx + C\]
\[\int sec^{2}x dx =tanx + C\]
\[\int cosecxcotx dx =-cosecx + C\]
\[\int secxtanxdx =secx+ C\]
\[\int\frac{1}{\sqrt{1-x^{2}}}dx=sin^{-1}x+C\]
\[\int\frac{-1}{\sqrt{1-x^{2}}}dx=cos^{-1}x+C\]
\[\int\frac{1}{1+x^{2}}dx=tan^{-1}x+C\]
\[\int\frac{-1}{1+x^{2}}dx=cot^{-1}x+C\]
\[\int a^{x}dx=\frac{a^{x}}{lna}+C\]
\[\int lnxdx=x[lnx-1]+C\]
\[\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}tan^{-1}\frac{x}{a}+C\]
\[\int\frac{1}{\sqrt{a^{2}-x^{2}}}dx=sin^{-1}\frac{x}{a}+C\]
Integration by Substitution
\[ If \int f(x)dx=F(x)+C\]
\[ then \int f(ax +b)dx=\frac{1}{a}F(ax+b)+C\]
\[\int\frac{f^{|}(x)}{f(x)}dx=ln|f(x)|+C\]
\[\int\frac{f^{|}(x)}{\sqrt{f(x)}}dx=2\sqrt{f(x)}+C\]
\[\int f^{||}(x).[f(x)]^{n}dx=\frac{[f(x)]^{n+1}}{n+1}+C\]
\[\int\frac{1}{a^{2}-x^{2}}dx=\frac{1}{2a}ln|\frac{a+x}{a-x}|+C\]
\[\int\frac{1}{x^{2}-a^{2}}dx=\frac{1}{2a}ln|\frac{x-a}{x+a}|+C\]
\[\int\frac{1}{\sqrt{x^{2}+a^{2}}}dx=\ln|x+ \sqrt{x^{2}+a^{2}} |+C\]
\[\int\frac{1}{\sqrt{x^{2}-a^{2}}}dx=\ln|x+ \sqrt{x^{2}-a^{2}} |+C\]
\[\int\sqrt{x^{2}+a^{2}}dx=\frac{1}{2}x\sqrt{x^{2}+a^{2}}+\frac{1}{2}a^{2}ln|x+ \sqrt{x^{2}+a^{2}} |+C\]
\[\int\sqrt{x^{2}-a^{2}}dx=\frac{1}{2}x\sqrt{x^{2}-a^{2}}-\frac{1}{2}a^{2}ln|x+ \sqrt{x^{2}-a^{2}} |+C\]
\[\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}x\sqrt{a^{2}-x^{2}}+\frac{1}{2}a^{2}sin^{-1}\frac{x}{a}+C\]
Put
\[\sqrt{a^{2}-x^{2}} , a^{2}-x^{2} \Rightarrow x=acos\theta , x=asin\theta \]
\[\sqrt{a^{2}-+x^{2}} , a^{2}+x^{2} \Rightarrow x=atan\theta , x=acot\theta \]
\[\sqrt{x^{2}-a^{2}} , a^{2}-x^{2} \Rightarrow x=acos\theta , x=asec\theta \]
\[\sqrt{a-x} , \sqrt{a+x} \Rightarrow x=acos2\theta \]
\[\sqrt{\frac{a-x}{a+x}} o, \sqrt{\frac{a+x}{a-x}} \Rightarrow x=acos2\theta \]
\[\sqrt{\frac{x-a}{b-x}} , \sqrt{(x-a)(b-x)} \Rightarrow x=acos^{2}\theta + bsin^{2}\theta \]
Derivatives
\[1+\frac{1}{x^{2}}=\frac{d}{dx} (x-\frac{1}{x})\]
\[1-\frac{1}{x^{2}}=\frac{d}{dx} (x+\frac{1}{x})\]
Integrals derived by Parts
\[\int f(x).g(x)dx=f(x) \int g(x)dx-\int f^{|}(x)[\int g(x)]dx+C\]
\[\int (f(x)g^{|}(x) + g(x)f^{|}(x))dx=g(x)f(x)+C\]
\[\int e^{x}(f(x)+f^{|}(x))dx=e^{x}f(x)+C\]
\[\int e^{g(x)}(f(x).g^{|}(x))+f^{|}(x))dx=e^{g(x)}f(x)+C\]
\[\int (f(x)+xf^{|}(x))dx=xf(x)+C\]
\[\int e^{ax}cosbxdx=\frac{e^{ax}}{a^{2}+b^{2}}[acosbx+bsinbx]+C\]
\[\int e^{ax}sinbxdx=\frac{e^{ax}}{a^{2}+b^{2}}[asinbx-bcosbx]+C\]
Integration of Algebric expression
For integrals of the type
\[\int\frac{p(x)}{g(x)}dx\]
\[\int\frac{p(x)}{\sqrt{g(x)}}dx\]
where \( p(x) , g(x) \) are quadratic polynomials, the numerator can be written as
\[ p(x)=\lambda_{1}g(x)+\lambda_{2}g^{|}(x)+\lambda_{3}\] or by using complete square method
If \( p(x) , g(x) \) are higher polynomials divide by suitable powers of x
Integration of Squared Algebric expression
\( P(x) \) is a linear or constant
Type1:
\[\int\frac{P(x)}{(px+q)\sqrt{ax+b}}dx\]
\[\int\frac{P(x)}{(px^{2}+qx+r)\sqrt{ax+b}}dx\]
Put \(ax+b=t^{2}\)
Type2:
\[\int\frac{P(x)}{(px+q)\sqrt{ax^{2}+bx+c}}dx\]
\[\int\frac{P(x)}{(px+q)^{r}\sqrt{ax^{2}+bx+c}}dx\]
Put \(px+q=\frac{1}{t}\)
Type3:
\[\int\frac{P(x)}{(px^{2}+q)\sqrt{ax^{2}+b}}dx\]
Put \(x=\frac{1}{t}\)
Integrals of Trigonometric functions
Type1:
\[\int\frac{1}{a+bcosx}dx\]
\[\int\frac{1}{a+bsinx}dx\]
\[\int\frac{1}{a+bcosx+csinx}dx\]
Put
\[cosx=\frac{1-tan^2\frac{x}{2}}{1+tan^{2}\frac{x}{2}}\]
\[sinx=\frac{2tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}\]
and obtain quadratic in terms of \[t=tan\frac{x}{2}\]
Type2:
\[\int\frac{sinx}{asinx+bcosx}dx\]
\[\int\frac{cosx}{asinx+bcosx}dx\]
\[\int\frac{pcosx+qsinx}{asinx+bcosx}dx\]
\[\int\frac{pcosx+qsinx+r}{asinx+bcosx+c}dx\]
For these write Numerator as like in algebric expressions
\[Nr=A(Dr)+B(Dr)^{|}+C\]
Type3:
\[\int\frac{1}{a+bcos^{2}x+csin^{2}x}dx\]
\[\int\frac{1}{asin^{2}x+bcos^{2}x}dx\]
\[\div \text{ by cos}^{2}x\]
Definite Integration
\[\text{If}\int f(x)dx = F(x)+ C\text{, then}\]
\[\int_{a}^{b}f(x)dx = F(b)-F(a) \]
Integral as a limit of sum
Properties of Definite Integration
Property1: \[\int_{a}^{b}f(x)dx=\int_{a}^{b}f(u)du\] Property2: \[\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\] Property3: \[\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx\]\[\text{ such that a < c < b}\] Property4: \[\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\] Property5: \[\int_{0}^{2a}f(x)dx=\int_{0}^{a}[f(x)+f(2a-x)]dx\] Property6: \[\int_{-a}^{+a}f(x)dx=2\int_{0}^{a}f(x)dx\]\[\text{ If f(x) is a even function}\] Property7: \[\int_{-a}^{+a}f(x)dx=0\]\[\text{ If f(x) is a odd function}\] Property8: \[\int_{0}^{2a}f(x)dx=2\int^{a}_{0}f(x)dx\] \[\text{If f(x)=f(2a-x)}\] Property9: \[\int_{0}^{2a}f(x)dx=0\] \[\text{If f(x)=-f(2a-x)}\] Property10: \[\int_{0}^{a}xf(x)dx=2a\int_{0}^{\frac{a}{2}} f(x)dx\] \[\text{If f(x)=f(a-x)}\] Property11: \[\int_{a}^{b}f(x)dx=(b-a)\int^{1}_{0}f[(b-a)x+a]dx\] \[\text{If f(x)=f(a-x)}\]
Definite Integration by using Parts
\[\int_{a}^{b}f(x)g(x)dx=|f(x)\int g(x)dx|_{a}^{b}-|\int f^{|}(x)(\int g(x))dx|_{a}^{b}\]Definite Integrals of Periodic functions
Property1: \[\int_{0}^{nT}f(x)dx=n\int_{0}^{T}f(x)dx\]
Property2: \[\int_{mT}^{nT}f(x)dx=(n-m)\int_{0}^{T}f(x)dx\]
Property3: \[\int_{a}^{a+nT}f(x)dx=n\int_{0}^{T}f(x)dx\]
Property4: \[\int_{a+nT}^{b+nT}f(x)dx=\int_{a}^{b}f(x)dx\]
\[\text{where m,n}\in\mathbb{Z}\]
Walli's Reduction formula
Type1:
\[\int_{0}^{\frac{\pi}{2}}sin^{n}xdx=\int_{0}^{\frac{\pi}{2}}cos^{n}xdx\]
\[ \int_{0}^{\frac{\pi}{2}}cos^{n}xdx = \begin{cases} \frac{(n-1).(n-3).(n-5)...2}{(n-2).(n-4).(n-6)...3} &; \text {when n is odd} \\ \frac{(n-1).(n-3).(n-5)...1}{(n-2).(n-4).(n-6)...2} \frac{\pi}{2}&; \text {when n is even} \end{cases} \]
Type2:
\[\text{If both m & n are odd }\mathbb{Z}^{+} \text{ or one of them } \mathbb{Z}^{+} \]
\[\int_{0}^{\frac{\pi}{2}}sin^{m}xcos^{n}xdx=\frac{(m-1).(m-3)....(n-1)(n-3)....}{(m+n)(m+n-2).....(1or2)}\]
Type3:
\[\text{If both m & n are even }\mathbb{Z}^{+}\]
\[\int_{0}^{\frac{\pi}{2}}sin^{m}xcos^{n}xdx=\frac{(m-1).(m-3)....(n-1)(n-3)....}{(m+n)(m+n-2).....(1or2)}\frac{\pi}{2}\]
Leibnitz Rule
\[\frac{d}{dx}(\int_{g(x)}^{h(x)}f(x)dx)=f(g(x))g^{|}(x)-f(h(x))h^{|}(x)\]
Determination of Area bounded by graphs
Geometrical Interpretation of Definite Integrals:
Consider \[f(x)=sinx\]
\[where \int_{0}^{\frac{3\pi}{2}} sinxdx= A1-A2+A3\]
Therefore, definite integration of algebric sum of area bounded by graphs
Determination of Area
Type1:
\[A=\int_{a}^{b}f(x)dx\]
\[\text{where f(x)>0 }\forall \text{ }x\text{ }\in\text{(a,b)}\]
Type2:
\[A=-\int_{a}^{b}f(x)dx\]
\[\text{where f(x)<0 }\forall \text{ }x\text{ }\in\text{(a,b)}\]
Type3:
\[A=\int_{a}^{b}f(x)dx-\int_{b}^{c}f(x)dx\]
\[\text{where f(b)=0; f(x)<0 }\forall \text{ }x \text{ }\in\text{(b,c)}\]
Type4:
\[A=\int_{a}^{b}(f(x)-g(x))dx\]
\[\text{where f(x)>g(x) }\forall \text{ }x\text{ }\in\text{(a,b)}\]
Type5:
\[A=\int_{a}^{b}f(y)dy\]
\[\text{where f(y)>0 }\forall \text{ }y\text{ }\in\text{(a,b)}\]
Type6:
\[A=-\int_{a}^{b}f(y)dy\]
\[\text{where f(y)<0 }\forall \text{ }y\text{ }\in\text{(a,b)}\]
Type7:
\[A=\int_{a}^{b}f(y)dy-\int_{b}^{c}f(y)dy\]
\[\text{where f(b)=0; f(x)<0 }\forall \text{ }y\text{ }\in\text{(b,c)}\]
Type8:
\[A=\int_{a}^{b}(f(y)-g(y))dy\]
\[\text{where f(y)>g(y) }\forall \text{ }y\text{ }\in\text{(a,b)}\]
Important Results
Differential Equation
Order & Degree of a Differential Equation
Order: The order of a differential equation is the highest order derivative present in the differential
equation.
Degree: If a differential equation is expressible in a polynomial form, then the integral power of the
highest order derivative appears is called the degree of the differential equation
\[1+\frac{dy}{dx}=y^{2}[\frac{d^{3}y}{dx^{3}}]^{3}\]
The order and degree of above equation is 3,2 respectively
\[e^{\frac{dy}{dx}}+sin\frac{d^{3}y}{dx^{3}}=0\]
The order of the above equation is 3 while degree is undefined as it cannot be expressed as a polynomial
Formation of a Differential Equation
A differential equation relates various parameters x, y, z which is satisfied a family of curves. If a family of curves has n arbitrary constant, then differentiate it n and eliminate arbitrary constant. Therefore, order of a DE is equal to number of arbitrary constant
Solution of a Differential Equation
General Solution: Solution of Differential Equation involving exactly the same number of arbitrary constant as in order of DE
Particular Solution: Solution of Differential Equation obtained by assigning values to arbitrary constant in a general solution
Methods of solving Differential Equation
Variable seperable method
Type1:
\[for \text{ }\frac{dy}{dx}=f(x)\]
\[\int f(x)dx=\int dy\]
Type2:
\[for \text{ }\frac{dx}{dy}=f(x)\]
\[\int \frac{1}{f(x)}dx=\int dy\]
Type3:
\[for \text{ }f(x)dx+g(y)dy=0\]
\[\int f(x)dx=-\int g(y)dy\]
Type4:
\[for \text{ }f(x)dy+g(y)dx=0\]
\[\int \frac{1}{g(y)}dy=-\int \frac{1}{f(x)}dx\]
Type5:
\[for \text{ }\frac{dy}{dx}=f(ax+by+c)\]
\[ ax+by+c=v;\text{ then diff ; } 1+\frac{dy}{dx}=\frac{dv}{dx}\]
\[\text{Put }\frac{dy}{dx}=\frac{dv}{dx}-1\]
Homogenous Equations
These are functions of the form
\[f(tx,ty)=t^{\lambda}f(x,y)\text{ }\]
Type1:
\[\frac{dy}{dx}=\frac{f(x,y)}{g(x,y)}\]
\[\text{ }\frac{y}{x}=v\text{; then diff }\]
\[\text{ Put }\frac{dy}{dx}=v+x\frac{dv}{dx}\]
Type2:
\[\frac{dx}{dy}=\frac{f(x,y)}{g(x,y)}\]
\[\text{ }\frac{x}{y}=v\text{; then diff }\]
\[\text{ Put }\frac{dx}{dy}=v+x\frac{dv}{dy}\]
Type3:
\[\frac{dy}{dx}=\frac{ax+by+c}{px+qy+r}\]
\[x-h=X\text{ ; }y-k=Y\]
\[\text{ Put }\frac{dY}{dX}=\frac{dy}{dx}\]
\[where\text{ }(h,k)\text{ is intersection of lines}\]
\[ax+by+c=o\text{ ; }px+qy+r=0\]
\[\frac{dY}{dX}=\frac{aX+bY}{pX+qY}\]
The above equation is of type 1 Homogenous equation
Linear Differential Equation
This type of equation has its dependent variable and its derivative appears to be in first degree.
Type1:
\[\frac{dy}{dx}+Py=Q\]
\[\text{ ; }where\text{ }P\text{ ; }Q\text{ are functions of x}\]
\[I.F=e^{\int Pdx}\text{ (Integrating factor) }\]
\[ye^{\int Pdx}=\int Qe^{\int Pdx}dx\]
Type2:
\[\frac{dx}{dy}+Px=Q\]
\[\text{ ; }where\text{ }P\text{ ; }Q\text{ are functions of y}\]
\[I.F=e^{\int Pdy}\text{ (Integrating factor) }\]
\[xe^{\int Pdy}=\int Qe^{\int Pdy}dy\]
Type3:
\[\frac{dy}{dx}+Py=Qy^{n}\]
\[\div\text{ above equation by }y^{n}\]
\[\frac{1}{y^{n}}\frac{dy}{dx}+P\frac{1}{y^{n-1}}=Q\]
\[Put \text{ }\frac{1}{y^{n-1}}=t;\text{ then diff}\]
\[\frac{1}{y^{n}}\frac{dy}{dx}=\frac{-1}{n-1}\frac{dt}{dx}\]
\[y^{1-n}e^{(1-n)\int Pdx}=(1-n)\int Qe^{(1-n)\int Pdx}dx\]
\[\frac{1}{n-1}\frac{dt}{dx}+Pt=Q\]
\[\frac{dt}{dx}+P't=Q'\]
\[where\text{ }P'=(n-1)P;\text{ }Q'=(n-1)Q\]
The above equation is of type 1 Linear Differential equation
Exact forms of Differential Equation
\[xdy+ydx=d(xy)\]
\[\frac{xdy+ydx}{xy}=d(lnxy)\]
\[\frac{dy+dx}{x+y}=d(ln(x+y))\]
\[\frac{xdy+ydx}{(xy)^{2}}=d(-\frac{1}{xy})\]
\[\frac{xdy+ydx}{x^{2}+y^{2}}=\frac{1}{2}d(ln(x^{2}+y^{2}))\]
\[\frac{ydx-xdy}{(y)^{2}}=d(\frac{x}{y})\]
\[\frac{ydx-xdy}{(xy)}=d(ln(\frac{x}{y}))\]
\[\frac{ydx-xdy}{(x^{2}+y^{2})}=d(tan^{-1}(\frac{x}{y})\]
\[\frac{ye^{x}dx-e^{x}dy}{(y^{2})}=d(\frac{e^{x}}{y})\]