solution of homogeneous differential equation

solution of homogeneous differential equation

prove that [ x^2 - y^2 ]=c [ x^2 - y^2 ]^2 is a solution of
[ x^3 - 3x (y^2) ] dx = [ y^3 - 3x^2y ] dy

first solve for (dy/dx)
take the function as f(x,y)
now check if the equation is a homogeneous differential equation
replace x and y with tx and ty
simplify and see if the t can be cancelled off to get back f(x,y)
If the equation is a homogeneous differential equation, put y=Vx
(dy/dx) = V + x(dV/dx)
simplify and separate the variables and then integrate

 

Variable separable differential equation

*find the equation of the curve passing through (0,pi/4) whose differential equation is
sinx cosy dx +  cosx siny dy =0
solution of  variable separable differential equation find the equation of the curve passing through (0,pi/4) whose differential equation issinx cosy dx +  cosx siny dy =0

* show that the general solution of y' + {[ (y^2)+y + 1]/[x^2+x+1] = 0
is given by x+y+1=A[1-x-y-2xy]

solution of variable separable differential equation from ncert cbse miscellaneous problem

*solve y e^(x/y) = [ y e^(x/y) + (y^2) ]dy  [ y is not equal to 0]
solution of differential equation which is not homogeneous but which can be solved using x=vy

*linear differential equation 

solve [ { e^[-2sqrt(x)] / sqrt(x) } - {y / sqrt(x)}] [dx/dy] = 1  [x is not 0]
solution of linear differential equation of ncert cbse miscellaneous differential equation

*linear differential equation of the type
solve y dx + [x-(y^2)]dy = 0
solution of linear differential equation of the type [dx/dy] + Px = Q


*find a particular solution of the differential equation [dy/dx]+ycotx =4xcosecx [x is not 0] given that y =0 if x=pi/2
solution of linear differential equation from ncert cbse miscellaneous  [dy/dx]+ycotx =4xcosecx [x is not 0] given that y =0 if x=pi/2

solution of a second order differential equation using reduction of order
solve y"-y = 0 if y = coshx is one of the solutions
using the formula for reduction of order
solution of solution of a second order differential equation using reduction of order

variation of parameter method

solve xy" - 4y' = x^4 by method of variation of parameter
solution to problem on differential questions using variation of parameter method

orthogonal trajectory of y(1+x ² ) = Cx

find the orthogonal trajectory of y(1+x ² ) = Cx
answer to problem on  orthogonal trajectory of y(1+x ² ) = Cx

orthogonal trajectory of y = (k/x)

find the orthogonal trajectory of y = (k/x)
solution to  find the orthogonal trajectory of y = (k/x)



formulae on integration
 
PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

 PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR


PAGE 6 INTEGRATION USING PARTIAL FRACTIONS

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