Type – 7. LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER:
The most general form of a linear differential equations of first order is (dy/dx) + Py =Q, where P & Q are functions of x. (Independent variable) In physics it will look like (ds/dt) + f(t) .s =g(t)
To solve such an equation multiply both sides by e ∫ Pdx.
Note: (1) The factor e ∫ Pdx on multiplying by which the left hand side of the differential equation becomes the differential coefficient of some function of x & y, is called integrating factor of the differential equation popularly abbreviated as I.F.
(2) It is very important to remember that on multiplying by the integrating factor, the left hand side becomes the derivate of the product of y and the I.F.
(3) Some times a given differential equation becomes linear if we take y as the independent variable and x as the dependent variable. e.g. the equation ;
(x + y + 1) (dy/dx) = y2 + 3 can be written as (y2 + 3) (dx/dy) =x +y +1 which is a linear differential equation.
Examples:
(i) t (1+t2) dx = (x + xt2 – t2) dt ; x|t=1 = – (p/4)
(ii)
(iii)
(iv) (x2 – 1) sinx (dy/dx) + [2x sinx + (x2 – 1)cosx ] y – (x2 – 1 ) cosx = 0
[Solution:
(v) Find the curve such that the initial ordinate of any tangent is less than the abscissa of the point of tangency by two units. [Ans: y = cx – x ln |x| – 2 ]
(vi) Find the curve such that the area of the rectangle constructed on the abscissa of any point and the initial ordinate of the tangent at this point is 4. (or a2 )
[Ans: y = cx ± (a2/2x) ]
[ Hint: D.E. derived for solution is |xy - x2(dy/dx) | =a2]
(vii) Y sinx (dy/dx) = cos x (sin x –y2) [ change of variable by a suitable substitution
put y2 = z 2y (dy/dx) =(dz/dx)
sinx (dz/dx) = 2cos x (sin x –z )
(dz/dx) = 2cos x – 2 cot x z = 2cos x which is linear
(viii)
(dx/dy) cosy.x=sin 2y
(xi) Prove the identity
deriving for the function I (x) = 0 ∫x eZx-Z2 a D.E. and solving it.
Diff. (1) with respect to x.
(x) Show that the solution of the D.E.
(1–x2) (dy/dx) + xy = ax are ellipses or hyperbolas with the centres at the point (0,a) and the axis parallel to the co-ordinate axes, each curve having one constant axis whose length is equal to 2.
