Fourier Series f(x) = $a_o/2 + ∑↙{n =1 }↖∞ (a_n cos nx + b_n sin nx)$ $a_o, a_n, b_n$ are fourier coefficients Note (Useful Results) (i) $∫↙{c}↖{c + 2π} cosnx dx = [{sin nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0) (ii) $∫↙{c}↖{c + 2π} sinnx dx = [-{cos nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0) (iii) $\table∫↙{c}↖{c + 2π} sin mx cosnx dx,=, 1/2 ∫↙{c}↖{c + 2π} [sin(m + n)x + sin(m-n) x] dx;,=,1/2 [- {cos(m+n)x}/{(m+n)} – {cos (m-n)x}/{(m-n)}]_{c}^{c + 2π} = 0 (m≠n) $ If m=n, $∫↙{c}↖{c + 2π} sinmx cosnx dx = 1/2 ∫↙{c}↖{c + 2π} ...
Read more
Rule to find the equation of orthogonal trajectories Note: Differentiate as many times as there are constants. For rectangular cartesian co-ordinates: Step 1: Given f(x,y,a) = 0, where a is a variable parameter. Step 2: Differentiate f(x,y,a) = 0 w.r.t. x and eliminate ‘a’. Step 3: Replace ${dy}/{dx}$ by $- {dx}/{dy}$ For polar co-ordinates: Step 1: Given f(r,θ,a) = 0, where a is a variable parameter. Step 2: Differentiate f(r,θ,${dr}/{dθ}$) = 0 w.r.t. x and eliminate ‘a’. Step 3: Replace ${dr}/{dθ}$ by ($- r^2{dθ}/{dr}$) Rate of decay of radioactive materials If u is amount of material at any time t, ...
Read more
Order of differential equation Definition: The order of differential equation is the order of highest derivation that appears in the equation Degree of differential equation Definition: It is the degree of highest order differential coefficient or derivative, when differential coefficients are free from fractions General solution of differential equation A relation between the dependent and independent variables, which is free from derivatives, which satisfies a given differential equation which contains arbitrary constants equal to the order of the differential equation is called general solution or complete integral. Particular solution of a differential equation The solution obtained by assigning particular values ...
Read more
List of Formulas (1) $∫ x^n dx = {x^{n+1}}/{n + 1} + c$ (2) $∫ 1/x dx = log|x| + c$ (3) $∫a^x dx = {a^x}/{loga} + c$ (4) $∫e^x dx = e^x + c$ (5) $∫sinx dx = – cosx + c$ (6) $∫cosx dx = sinx + c$ (7) $∫tanx dx = log |secx| + c$ (8) $∫cotx dx = log |sinx| + c$ (9) $∫secx dx = log |secx + tanx| + c$ (10) $∫cosecx dx = log |cosecx – cotx| + c$ (11) $∫sec^2 x dx = tanx + c$ (12) $∫cosec^2 x = – cotx ...
Read more