Ordinary Differential Equations

Ordinary Differential Equations

1. First Order Equations

   • Variable Separable
     • Seperate the x function and dx on one side and y variable function on other side and integrate it
   • Homogeneous
     • use substitution V=FX then it will become variable separable
     • Equation of the form : (a1x + b1y + c1) + (a2x + b2y + c2) + 0
       • Reduce it to homogeneous
         • When a1/a2 = b1/b2
           • use substitution x = X + h and y= Y + k.... put constants = 0.... then find the value of h and k ... remmaining eq. will be homogeneous... use Y=VX ... it will become variable separable .... then in solution put the value of use substitution V
         • When a1/a2 =/= b1/b2
           • use substitution a1x + b1y = z .... bring the equation in x and z form or y and z form then it will become homogeneous
   • Linear Equations
     • eq. of the form dy/dx + p(x)y = q(x)
       • find integrating factor i.e ∫ e^p(c) dx
         • multiply it with equation ... then we will get product formula on right side.... integrate it and we will get solution
   • Exact Equations
     • eq. of the form M(x,y)dx + N(x,y)dy = 0
       • when ∂M/∂y = ∂N/∂x
         • use .... ∫M(x,y)dx + ∫ (terms in N free from x )dy = c
       • when ∂M/∂y =/= ∂N/∂x
         • My - Nx / N(x,y) find integrating factor ... multiply it with eq. ... it will become an exact eq.
         • Nx - My / M(x,y) find integrating factor ... multiply it with eq. ... it will become an exact eq.
           • the less no. of terms will be in deniminator .....
   • Bernoulli Equation
     • eq. of the form dy/dx + p(x)y = q(x)(y^n)
       • we will solve it by converting it into linear form
         • first divide by y^n .... put v = y^1-n ..... fing dv/dx .... put the value of v and dv/dx in eq. ..... then multiply with (1-n) ... it will be linear ....

2. Second Order Equations

   • case 1
     • When there is 0 on right side ... convert in D operator form... convert it into auxilliary form .... find roots
       • When roots are real and same
         • Use (A+Bx)e^@x
       • when roots are real and distinct
         • Use (Ae^r1x)+Be^r2x
       • when roots are imaginary
         • e^∝x(Acosβx + Bsinβx)
   • Case 2
     • when there is e^∝x on the right side ... find yc as before ... Yp = [1/f(D)]*e^∝x.... replace D by ∝ and solve it
   • Case 3
     • when there is X^n on the right side ... find yc as before ... Yp = [1/f(D)]*X^n ... use binomial series and find solution
   • Case 4
     • when there is cos∝x / sin∝x on the right side ... find yc as before ... Yp = [1/f(D)]*cos∝x ..Replace D^2 by (-∝^ 2) if we do have D in dinominator . we will rationalize it and find the solution.
   • Case 5
     • when there is (e^∝x)(x^n) on the right side find yc as before ... replace D by D + ∝ then use binomial series than operate on X^n
   • Case 6
     • When there is (e^∝x)(cosβx) on the right side .find yc as before.. replace D by D+∝ ... replace D^2 by -β^2 ans solve it

3. Degree of equation

   • power of highest order derivative

4. order of equation

   • highest order derivative

5. how to identify equations....

   • Linear Equations ....
     • degree should be 1 ...... there should be no product of y and its derivatives ....... if another y or its derivative is present , their power should be 1 or 0
   • Homogeneous Equations
     • if the power of variables is same ....
       • how to identify it...
         • first order
           • put x = xt and y = yt .... if the same eq. wil return then it will be homogeneous
         • higher order
           • if each term is either the derivative of dependent variable then eq. is homogeneous

6. interval of validity

   • the interval in which solution of eq. lies

7. solution of differential eq.s are called integral curves

8. formation of differential eq.s

   • if there are n no. of constants in eq. then diffrentiate it n times ... then solve all resulted eq.s simultaneously to get DE