EXERCISES

1. Suppose that*z*_{1}is a solution of*L*(*y*) =*d*_{1} and that*z*_{2}is a solution of*L*(*y*) =*d*_{2}. Then
show that*z*_{1}+*z*_{2} is a solution of the equation

*L*(*y*(*t*)) =*d*_{1}(*t*) +*d*_{2}(*t*)*.*

2. If a complex valued function*z* is a solution of the equation*L*(*x*) = 0 then, show that
the real and imaginary parts of*z* are also solutions of*L*(*x*) = 0*.*

3. (Reduction of the order) Consider an equation

*L*(*x*) =*a*_{0}(*t*)*x*^{00}+*a*_{1}(*t*)*x*^{0}+*a*_{2}(*t*)*x*= 0*,* *a*_{0}(*t*)*6*= 0*, t∈I*.

where*a*_{0}*, a*_{1} and*a*_{2} are continuous functions defined on*I*. Let*x*_{1}*6*= 0 be a solution of
this equation. Show that*x*_{2} defined by

*x*_{2}(*t*) =*x*_{1}(*t*)
Z _{t}

*t*0

1
*x*^{2}_{1}(*s*)exp

³

*−*
Z _{s}

*t*0

*a*_{1}(*u*)
*a*_{0}(*u*)*du*

´

*ds,* *t*_{0} *∈I,*

is also a solution. In addition, show that*x*_{1} and *x*_{2} are linearly independent on*I*.

*x*^{0}_{p} =*x*^{0}_{1}*u*_{1}+*x*^{0}_{2}*u*_{2}+ (*x*_{1}*u*^{0}_{1}+*x*_{2}*u*^{0}_{2}).

We do not wish to end up with second order equations for *u*_{1}*, u*_{2} and naturally we choose
*u*_{1} and *u*_{2} to satisfy

*x*_{1}(*t*)*u*^{0}_{1}(*t*) +*x*_{2}(*t*)*u*^{0}_{2}(*t*) = 0 (2.17)
Added to it, we already known how to solve first order equations. With (2.17) in hand we
now have

*x*^{0}_{p}(*t*) =*x*^{0}_{1}(*t*)*u*_{1}(*t*) +*x*^{0}_{2}(*t*)*u*_{2}(*t*)*.* (2.18)
Differentiation of (2.18) leads to

*x*^{00}_{p} =*u*^{0}_{1}*x*^{0}_{1}+*u*_{1}*x*^{00}_{1}+*u*^{0}_{2}*x*^{0}_{2}+*u*_{2}*x*^{00}_{2}*.* (2.19)
Now we substitute (2.16)*,*(2.18) and (2.19) in (2.14) to get

[*a*_{0}(*t*)*x*^{00}_{1}(*t*) +*a*_{1}(*t*)*x*^{0}_{1}(*t*) +*a*_{2}(*t*)*x*_{1}(*t*)]*u*_{1}+ [*a*_{0}(*t*)*x*^{00}_{2}(*t*) +*a*_{1}(*t*)*x*^{0}_{2}(*t*) +*a*_{2}(*t*)*x*_{2}(*t*)]*u*_{2}+
*u*^{0}_{1}*a*_{0}(*t*)*x*^{0}_{1}+*u*^{0}_{2}*a*_{0}(*t*)*x*^{0}_{2} =*d*(*t*)*,*

and since *x*_{1} and *x*_{2} are solutions of (2.15), hence

*x*^{0}_{1}*u*^{0}_{1}(*t*) +*x*^{0}_{2}*u*^{0}_{2}(*t*) = *d*(*t*)

*a*_{0}(*t*)*.* (2.20)

We solve for*u*^{0}_{1} and *u*^{0}_{2} from (2.17) and (2.20), to determine*x*_{p}*.*It is easy to see
*u*^{0}_{1}(*t*) = _{a} ^{−x}^{2}^{(t)d(t)}

0(*t*)*W*[*x*1(*t*)*,x*2(*t*)]

*u*^{0}_{2}(*t*) = _{a}_{0}_{(t)W[x}^{x}^{1}^{(t)d(t)}_{1}_{(t),x}_{2}_{(t)]}

where*W*[*x*_{1}(*t*)*, x*_{2}(*t*)] is the Wronskian of the solutions*x*_{1}and*x*_{2}. Thus,*u*_{1} and*u*_{2} are given
by

*u*_{1}(*t*) =*−*R _{x}

2(*t*)*d*(*t*)
*a*0(*t*)*W*[*x*1(*t*)*,x*2(*t*)]*dt*
*u*_{2}(*t*) =R _{x}_{1}_{(t)d(t)}

*a*0(*t*)*W*[*x*1(*t*)*,x*2(*t*)]*dt*

(2.21)

Now substituting the values of *u*_{1} and *u*_{2} in (2.16) we get a desired particular solution of
the equation (2.14). Indeed

*x*_{p}(*t*) =*u*_{1}(*t*)*x*_{1}(*t*) +*u*_{2}(*t*)*x*_{2}(*t*)*,* *t∈I*
is completely known. To conclude, we have :

Theorem 2.4.1. *Let the functions* *a*_{0}*, a*_{1}*, a*_{2} *and* *d* *in* (2.14) *be continuous functions on* *I.*

*Further assume that* *x*_{1} *and* *x*_{2} *are two linearly independent solutions of* (2.15)*. Then, a*
*particular solution* *x*_{p} *of the equation* (2.14)*is given by* (2.16)*.*

Theorem 2.4.2. *The general solution* *x*(*t*) *of the equation*(2.14) *onI* *is*
*x*(*t*) =*x*_{p}(*t*) +*x*_{h}(*t*)*,*

*where* *x*_{p} *is a particular solution given by* (2.16)*and* *x*_{h} *is the general solution of* *L*(*x*) = 0*.*

Also, we note that we have an explicit expression for *x*_{p} which was not so while proving
Theorem 2*.*3*.*15. The following example is for illustration.

### Lecture 11

Example 2.4.3. Consider the equation

*x*^{00}*−*^{2}_{t}*x*^{0}+_{t}^{2}2*x*=*t*sin*t,* *t∈*[1*,∞*).

Note that *x*_{1} = *t* and *x*_{2} = *t*^{2} are two linearly independent solutions of the homogeneous
equation on [1*,∞*). Now

*W*[*x*_{1}(*t*)*, x*_{2}(*t*)] =*t*^{2}.

Substituting the values of *x*_{1}*, x*_{2}*, W*[*x*_{1}(*t*)*, x*_{2}(*t*)]*, d*(*t*) = *t*sin*t* and *a*_{0}(*t*) *≡* 1 in (2.21), we
have

*u*_{1}(*t*) =*t*cos*t−*sin*t*
*u*_{2}(*t*) = cos*t*

and the particular solution is*x*_{p}(*t*) =*−t*sin*t*. Thus, the general solution is
*x*(*t*) =*−t*sin*t*+*c*_{1}*t*+*c*_{2}*t*^{2},

where*c*_{1} and *c*_{2} are arbitrary constants.

The method of variation of parameters has an extension to equations of order*n*(*n >* 2)
which we state in the form of a theorem, the proof of which has been omitted. Let us
consider an equation of the*n*-th order

*L*(*x*(*t*)) =*a*_{0}(*t*)*x*^{n}(*t*) +*a*_{1}(*t*)*x*^{n−1}(*t*) +*· · ·*+*a*_{n}(*t*)*x*(*t*) =*d*(*t*)*, t∈I.* (2.22)
Theorem 2.4.4. *Let* *a*_{0}*, a*_{1}*,· · ·* *, a*_{n}*, d*:*I* *→*R*be continuous functions. Let*

*c*_{1}*x*_{1}+*c*_{2}*x*_{2}+*· · ·*+*c*_{n}*x*_{n}

*be the general solution ofL*(*x*) = 0*. Then, a particular solution* *x*_{p} *of* (2.22) *is given by*
*x*_{p}(*t*) =*u*_{1}(*t*)*x*_{1}(*t*) +*u*_{2}(*t*)*x*_{2}(*t*) +*· · ·*+*u*_{n}(*t*)*x*_{n}(*t*)*,*

*where* *u*_{1}*, u*_{,}*· · ·* *, u*_{n} *satisfy the equations*

*u*^{0}_{1}(*t*)*x*_{1}(*t*) +*u*^{0}_{2}(*t*)*x*_{2}(*t*) +*· · ·*+*u*^{0}_{n}(*t*)*x*_{n}(*t*) = 0
*u*^{0}_{1}(*t*)*x*^{0}_{1}(*t*) +*u*^{0}_{2}(*t*)*x*^{0}_{2}(*t*) +*· · ·*+*u*^{0}_{n}(*t*)*x*^{0}_{n}(*t*) = 0

*· · · ·*
*u*^{0}_{1}(*t*)*x*^{(n−2)}_{1} (*t*) +*u*^{0}_{2}(*t*)*x*^{(n−2)}_{2} (*t*) +*· · ·*+*u*^{0}_{n}(*t*)*x*^{(n−2)}_{n} (*t*) = 0
*a*_{0}(*t*)£

*u*^{0}_{1}(*t*)*x*^{(n−1)}_{1} (*t*) +*u*^{0}_{2}(*t*)*x*^{(n−1)}_{2} (*t*) +*· · ·*+*u*^{0}_{n}(*t*)*x*^{(n−1)}_{n} (*t*)¤

=*d*(*t*)*.*

The proof of the Theorem 2.4.4 is similar to the previous one with obvious modifications.

EXERCISES

1. Find the general solution of*x*^{000}+*x*^{00}+*x*^{0}+*x*= 1 given that cos*t,*sin*t*and*e*^{−t}are three
linearly independent solutions of the corresponding homogeneous equation. Also find
the solution when *x*(0) = 0*, x*^{0}(0) = 1*, x*^{00}(0) = 0.

2. Use the method of variation of parameter to find the general solution of*x*^{000}*−x*^{0} =*d*(*t*)
where

(i) *d*(*t*) =*t*, (ii) *d*(*t*) =*e*^{t}, (iii) *d*(*t*) = cos*t*, and (iv) *d*(*t*) =*e*^{−t}.

In all the above four problems assume that the general solution of *x*^{000} *−x*^{0} = 0 is
*c*_{1}+*c*_{2}*e*^{−t}+*c*_{3}*e*^{t}.

3. Assuming that cos*Rt* and ^{sinRt}_{R} form a linearly independent set of solutions of the
homogeneous part of the differential equation *x*^{00} +*R*^{2}*x* = *f*(*t*)*, R* *6*= 0*, t* *∈* [0*,∞*),
where *f*(*t*) is continuous for 0 *≤* *t <* *∞* show that a solution of the equation under
consideration is of the form

*x*(*t*) =*A*cos*Rt*+*B*

*R*sin*Rt*+ 1
*R*

Z _{t}

0

sin[*R*(*t−s*)]*f*(*s*)*ds,*

where *A* and *B* are some constants. Show that particular solution of (2.14) is not
unique. (Hint : If*x*_{p}is a particular solution of (2.14) and*x*is any solution of (2.15) then
show that *x*_{p}+*c x*is also a particular solution of (2.14) for any arbitrary constant*c*.)
Two Useful Formulae

Two formulae proved below are interesting in themselves. They are also useful while studying boundary value problems of second order equations. Consider an equation

*L*(*y*) =*a*_{0}(*t*)*y*^{00}+*a*_{1}(*t*)*y*^{0}+*a*_{2}(*t*)*y*= 0*,* *t∈I*,

where*a*_{0}*, a*_{1}*, a*_{2} :*I* *→*Rare continuous functions in addition*a*_{0}(*t*) *6*= 0 for*t∈I*. Let *u* and
*v* be any two twice differentiable functions on*I*. Consider

*uL*(*v*)*−vL*(*u*) =*a*_{0}(*uv*^{00}*−vu*^{00}) +*a*_{1}(*uv*^{0}*−vu*^{0})*.* (2.23)
The Wronskian of *u*and *v* is given by *W*(*u, v*) =*uv*^{0}*−vu*^{0} which shows that

*d*

*dtW*(*u, v*) =*uv*^{00}*−vu*^{00}.

Note that the coefficients of *a*_{0} and *a*_{1} in the relation (2.23) are *W*^{0}(*u, v*) and *W*(*u, v*)
respectively. Now we have

Theorem 2.4.5. *If* *u* *andv* *are twice differential functions onI, then*
*uL*(*v*)*−vL*(*u*) =*a*_{0}(*t*)*d*

*dtW*[*u, v*] +*a*_{1}(*t*)*W*[*u, v*]*,* (2.24)
*where* *L*(*x*) *is given by* (2.7)*. In particular, if* *L*(*u*) =*L*(*v*) = 0 *then* *W* *satisfies*

*a*_{0}*dW*

*dt* [*u, v*] +*a*_{1}*W*[*u, v*] = 0*.* (2.25)

Theorem 2.4.6. *(*Able’s Formula*) Ifuand* *vare solutions of* *L*(*x*) = 0*given by* (2.7)*, then*
*the Wronskian of* *u* *and* *v* *is given by*

*W*[*u, v*] =*k* exp
h

*−*

Z *a*_{1}(*t*)
*a*_{0}(*t*)*dt*

i
*,*
*where* *k* *is a constant.*

*Proof.* Since *u* and *v* are solutions of *L*(*y*) = 0, the Wronskian satisfies the first order
equation (2.25) and Solving we get

*W*[*u, v*] =*k*exp
h

*−*

Z *a*_{1}(*t*)
*a*_{0}(*t*)*dt*

i

(2.26)
where*k* is a constant.

The above two results are employed to obtain a particular solution of a non-homogeneous second order equation.

Example 2.4.7. Consider the general non-homogeneous initial value problem given by
*L*(*y*(*t*)) =*d*(*t*)*, y*(*t*_{0}) =*y*^{0}(*t*_{0}) = 0*, t, t*_{0}*∈I,* (2.27)
where*L*(*y*) is as given in (2.14). Assume that*x*_{1}and*x*_{2}are two linearly independent solution
of*L*(*y*) = 0. Let*x* denote a solution of*L*(*y*) =*d*. Replace*u* and *v* in (2.24) by*x*_{1} and *x* to
get

*d*

*dtW*[*x*_{1}*, x*] +*a*_{1}(*t*)

*a*_{0}(*t*)*W*[*x*_{1}*, x*] =*x*_{1} *d*(*t*)

*a*_{0}(*t*) (2.28)

which is a first order equation for*W*[*x*_{1}*, x*]. Hence
*W*[*x*_{1}*, x*] = exp

h

*−*
Z _{t}

*t*0

*a*_{1}(*s*)
*a*_{0}(*s*)*ds*

i Z *t*
*t*0

exp£ R_{s}

*t*0

*a*1(*u*)
*a*0(*u*)*du*¤

*x*_{1}(*s*)*ds*

*a*_{0}(*s*) *ds* (2.29)

While deriving (2.29) we have used the initial conditions*x*(*t*_{0}) =*x*^{0}(*t*_{0}) = 0 in view of which
*W*[*x*_{1}(*t*_{0})*, x*(*t*_{0})] = 0. Now using the Able’s formula, we get

*x*_{1}*x*^{0}*−xx*^{0}_{1} =*W*[*x*_{1}*, x*_{2}]
Z _{t}

*t*0

*x*_{1}(*s*)*d*(*s*)

*a*_{0}(*s*)*W*[*x*_{1}(*s*)*, x*_{2}(*s*)]*ds.* (2.30)
The equation (2.30) as well could have been derived with*x*_{2} in place of *x*_{1} in order to get

*x*_{2}*x*^{0}*−xx*^{0}_{2} =*W*[*x*_{1}*, x*_{2}]
Z _{t}

*t*0

*x*_{2}(*s*)*d*(*s*)

*a*_{0}(*s*)*W*[*x*_{1}(*s*)*, x*_{2}(*s*)]*ds.* (2.31)
From (2.30) and (2.31) one easily obtains

*x*(*t*) =
Z _{t}

*t*0

£*x*_{2}(*t*)*x*_{1}(*s*)*−x*_{2}(*s*)*x*_{1}(*t*)¤
*d*(*s*)

*a*_{0}(*s*)*W*[*x*_{1}(*s*)*, x*_{2}(*s*)] *ds.* (2.32)
It is time for us to recall that a particular solution in the form of (2.32) has already been
derived while discussing the method of variation of parameters.