5.Solve the following initial value problems
(i) y′− 3 y=e^5 x y( 0 )= 0
(ii) xy′− 2 y=x^2 y( 1 )= 2
(iii) xy′+ 2 y=x^2 y( 1 )= 0(iv) xy′+ 3 y=sinx
x^2y(π)= 06.Solve the following second order equations
(i) y′′+y′− 2 y= 0 (ii) y′′− 4 y′+ 4 y= 0
(iii) y′′+ 4 y′+ 5 y=0(iv)y′′+ 4 y= 0
(v) y′′− 9 y=0(vi)y′′+y= 07.Obtain the general solution of the inhomogeneous equations formed by adding the
following right-hand sides to each of the equations of Q6.
(a) 2 (b) x+1(c)e−^2 x
(d) 2 sinx8.Solve the initial value problemy( 0 )=0,y′( 0 )=0 for each of the equations solved in
Q7(a). If you feel really keen press on with the other questions – you can check your
answers by substituting into the equation.
9.Solve the boundary value problemy( 0 )=y( 1 )=0 for each of the equations solved
in Q7(a). Again press on with the rest of Q7 if you need more practice.
15.8 Applications
There are a number of well known applications of first order equations which provide
classic prototypes for mathematical modelling. These mainly rely on the interpretation of
dy/dtas a rate of change of a functionywith respect to timet. In everyday life there are
many examples of the importance of rates of change – speed of moving particles, growth
and decay of populations and materials, heat flow, fluid flow, and so on. In each case we
can construct models of varying degrees of sophistication to describe given situations.
1.The rate at which a radioactive substance decays is found to be proportional to the
massm(t)present at timet. Write down and solve the differential equation expressing
this relation, given that the initial mass ism( 0 )=m 0 and whent=1,m=
m 0
2.- Newton’s law of coolingstates that the rate at which an object cools is proportional
to the difference between the temperature at the surface of the body, and the ambient
air temperature. Thus, ifT is the surface temperature at timetandTais the ambient
temperature, then
dT
dt=−λ(T−Ta)T( 0 )=T 0