1729 SANGAM

 REDUCTION FORMULA ∫(x^n e^ax )dx      ∫( x^2 e^(-x) ) dx ∫  cosⁿ(x)  dx   ∫ cos ^5⁡x  dx   ∫ cos ^m⁡ x.  sin ⁡ nx dx ∫  x^n sin⁡x dx  

D'Alembet's solution for one-dimensional wave equation ( homogeneous) Consider the initial value formula  ( ¶ ^2 u )/( ¶ x^2 ) =1/c^2    ( ¶ ^2 u)/( ¶ t^2 )    - ∞<x<∞    t>0 Satisfying the initial conditions u (x,0)=f(x) ,  ∂u/∂t (x,0)=g(x) This problem is known as the Cauchy problem for  one-dimensional  wave  equation.   D’Alembert’s  solution for  one-dimensional wave equation (non-homogeneous ) Consider one-dimensional wave equation   u_tt-c^2 u_xx=F(x,t) - ∞<x<∞    t>0  .  u(x,0)=f(x) ,  u_t  ( x,0)=g(x)   - ∞<x<∞ Where F ( x,t)  is the source acting on an incoming string 

  ( x - y ) dx -( x + y ) dy =0 Exact Equation If the first order partial derivatives of M( x,y ) and N( x,y ) are continuous then M dx + N dy = 0 is an exact equation if and only if  ∂ M/ ∂ y = ∂ N/∂ x Hence to solve the exact equation  M dx + N dy = 0   Integrate M with respect to x keeping y constant Integrate those terms in N not containing  x  with respect to   y The sum of those two integrals equated to c is the solution. EXAMPLE 1 ( x^2-2xy+3y^2 )dx+(y^2+6xy-x^2 )dy=0 EXAMPLE 2 ( 2xy-sec^2⁡x )dx+(x^2+2y)dy=0 EXAMPLE 3 Find the value of ‘ n ’ for which ( x+ye^2xy )dx+nxe^2xy  dy=0    and solve it EXAMPLE 4 ( sin⁡ x. tan⁡ y+1)dx-(cos⁡x.sec^2⁡y )dy=0 EXAMPLE 5 ( e^y+1)  cos⁡ x  dx+e^y  sin⁡x  dy=0