TYPE - 6 EQUATIONS REDUCIBLE TO THE HOMOGENEOUS FORM:
If ; (dy/dx) =( a1x+b1y +c1) /(a2x + b2y +c2) ; where a1b2 – a2b1 ¹ 0, i.e.(a1/b1) ¹ (a2/b2)
then the substitution x = u + h, y = v + k transform this equation to a homogeneous type in the new variables u and v where h and k are arbitrary constant to be chosen so as to make the given equation homogeneous which can be solved by the method as given in Type – 3. If
(i) a1b2 – a2 b1 = 0, then a substitution u = a1x + b1y transforms the differential equation to an equation with variables separable and
(ii) b1 + a2 = 0, then a simple cross multiplication and substituting d(xy) for x dy + y dx & integrating term by term yields the result easily.
Consider
LINEAR DIFERNTIAL EQUATIONS :
A differnential equation is said to be linear if the dependent variable & all its differential coefficients occur in degree one only and are never multiplied together.
The nth order linear differential equation is of the form;
A0 (x) (dny/dx2) +a1 (x) (dn-1/dxn-1) + …. An (x) .y =Æ (x) Where a0(x), a1 (x) .........an (x)......a0(x) are called
the coefficients of the differential equation.
Note that a linear differential equation is always of the first degree but every differential equation of the first degree nedd not be linear. e.g. the differential equation (d2y/dx2) +(dy/dx)3 +y2 =0 is not linear, through its degree is 1.
