Rev_T01
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Transcript of Rev_T01
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7/21/2019 Rev_T01
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REVIEW T01 - MA1506
Andreas DMG ([email protected]) &Ganesh Swaminathan ([email protected])
DIFFERENTIAL EQUATION
ADifferential Equation (DE)is an equation that relates a function withits derivatives. Theorderof a differential equation is the highest order of thederivative terms that are involved. A DE is said to be linear, if it can be writtenin the form:
any(n)(x) +an1y
(n1)(x) +...+a0y(x) =F(x).
In general, a DE has more than one solution. The general solution of a
DE with order n will have n arbitrary constant; and any solution obtainedby substituting certain values to each arbitrary constant is called particularsolution.
Examples:
1. The DE x(x+ 1)y = 1 has order= 1.
2. The DE y = [3 + (y)2]2/3 has order= 3.
3. The DE y = 1 has general solution y = x+ c, where c is an arbitraryconstant; and y= x+ 1 is a particular solution for the DE.
SEPARABLE D.E. (T01 Q1, Q2, Q3, Q4, Q5)
A DE is said to be separableif it can be written as M(x)dx= N(y)dy. Inthis case, the general solution can be obtained by integrating both sides, i.e.,M(x)dx=
N(y)dy+C
REDUCTION TO SEPARABLE D.E. VIA CHANGE OF VARI-ABLE (T01 Q5)
Some DE that are not separable can be reduced into separable DE by thechange of variable method. There are two techniques related to this method:
1. For function that can be written as y = gyx
, set v = yx , then substitute
y = vx and y
=v + xv
(i.e. do the change of variable method to changeyinto v). Solve the new DE (in v and x), and y can be obtained by y = vx(Q5(c)).
2. For function that can be written as y = g(ax+by+c) with b = 0, setv = ax+by + c and do the change of variable method. Solve the new DE,and y can be obtained by y = vaxcb (Q5(a) and (b)).
Remark: Sometimes the DE is not in any of the above mentioned form (as inQ5(c)). In that case, we may need to try out combination of the above methodsor a new method.
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