Differential Equations: Solutions of Differential Equations

First-order equations. The validity of term-by-term differentiation of a power series within its interval of convergence implies that first-order differential equations may be solved by assuming a solution of the form

substituting this into the equation, and then determining the coefficients c_n .

Example 1: Find a power series solution of the form

for the differential equation

Substituting

into the differential equation yields

Now, write out the first few terms of each series,

and combine like terms:

Since the pattern is clear, this last equation may be written as

In order for this equation to hold true for all x, every coefficient on the left-hand side must be zero. This means c₁ = 0, and for all n ≥ 2,

This last equation defines the recurrence relation that holds for the coefficients of the power series solution:

Since there is no constraint on c₀, c₀ is an arbitrary constant, and it is already known that c₁ = 0. The recurrence relation above says c₂ = ½ c₀ and c₃ = ⅓ c₁, which equals 0 (because c₁ does). In fact, it is easy to see that every coefficient c_n with n odd will be zero. As for c₄, the recurrence relation says

and so on. Since all c_n with n odd equal 0, the desire power series solution is therefore

Note that the general solution contains one parameter ( c₀), as expected for a first-order differential equation. This power series is unusual in that it is possible to express it in terms of an elementary function. Observe:

It is easy to check that y = c₀ e^x2/² is indeed the solution of the given differential equation, y′ = xy. Remember: Most power series cannot be expressed in terms of familiar, elementary functions, so the final answer would be left in the form of a power series.

Example 2: Find a power series expansion for the solution of the IVP

Substituting

into the differential equation yields

or, collecting all the terms on one side,

Writing out the first few terms of the series yields

or, upon combining like terms,

Now that the pattern is clear, this last equation can be written

In order for this equation to hold true for all x, every coefficient on the left-hand side must be zero. This means

The last equation defines the recurrence relation that determines the coefficients of the power series solution:

The first equation in (*) says c₁ = c₀, and the second equation says c₂ = ½(1 + c₁) = ½(1 + c₀). Next, the recurrence relation says

and so on. Collecting all these results, the desired power series solution is therefore

Now, the initial condition is applied to evaluate the parameter c₀:

Therefore, the power series expansion for the solution of the given IVP is

If desired, it is possible to express this in terms of elementary functions. Since

equation (**) may be written

which does indeed satisfy the given IVP, as you can readily verify.

Second-order equations. The process of finding power series solutions of homogeneous second-order linear differential equations is more subtle than for first-order equations. Any homogeneous second-order linear differential equation may be written in the form

If both coefficient functions p and q are analytic at x₀, then x₀ is called an ordinary point of the differential equation. On the other hand, if even one of these functions fails to be analytic at x₀, then x₀ is called a singular point. Since the method for finding a solution that is a power series in x₀ is considerably more complicated if x₀ is a singular point, attention here will be restricted to power series solutions at ordinary points.

Example 3: Find a power series solution in x for the IVP

Substituting

into the differential equation yields

The solution may now proceed as in the examples above, writing out the first few terms of the series, collecting like terms, and then determining the constraints on the coefficients from the emerging pattern. Here's another method.

The first step is to re-index the series so that each one involves xⁿ . In the present case, only the first series must be subjected to this procedure. Replacing n by n + 2 in this series yields

Therefore, equation (*) becomes

The next step is to rewrite the left-hand side in terms of a single summation. The index n ranges from 0 to ∞ in the first and third series, but only from 1 to ∞ in the second. Since the common range of all the series is therefore 1 to ∞, the single summation which will help replace the left-hand side will range from 1 to ∞. Consequently, it is necessary to first write (**) as

and then combine the series into a single summation:

In order for this equation to hold true for all x, every coefficient on the left-hand side must be zero. This means 2 c₂ + c₀ = 0, and for n ≥ 1, the following recurrence relation holds:

Since there is no restriction on c₀ or c₁, these will be arbitrary, and the equation 2 c₂ + c₀ = 0 implies c₂ = −½ c₀. For the coefficients from c₃ on, the recurrence relation is needed:

The pattern here isn't too difficult to discern: c_n = 0 for all odd n ≥ 3, and for all even n ≥ 4,

This recurrence relation can be restated as follows: for all n ≥ 2,

The desired power series solution is therefore

As expected for a second-order differential equation, the general solution contains two parameters ( c₀ and c₁), which will be determined by the initial conditions. Since y(0) = 2, it is clear that c₀ = 2, and then, since y′(0) = 3, the value of c₁ must be 3. The solution of the given IVP is therefore

Example 4: Find a power series solution in x for the differential equation

Substituting

into the given equation yields

Now, all series but the first must be re-indexed so that each involves xⁿ :

Therefore, equation (*) becomes

The next step is to rewrite the left-hand side in terms of a single summation. The index n ranges from 0 to ∞ in the second and third series, but only from 2 to ∞ in the first and fourth. Since the common range of all the series is therefore 2 to ∞, the single summation which will help replace the left-hand side will range from 2 to ∞. It is therefore necessary to first write (**) as

and then combine the series into a single summation:

Again, in order for this equation to hold true for all x, every coefficient on the left-hand side must be zero. This means c₁ + 2 c₂ = 0, 2 c₂ + 6 c₃ = 0, and for n ≥ 2, the following recurrence relation holds:

Since there is no restriction on c₀ or c₁, these will be arbitrary; the equation c₁ + 2 c₂ = 0 implies c₂ = −½ c₁, and the equation 2 c₂ + 6 c₃ = 0 implies c₃ = −⅓ c₂ = −⅓(-½ c₁) =¹/₆ c₁. For the coefficients from c₄ on, the recurrence relation is needed:

The desired power series solution is therefore

Determining a specific pattern to these coefficients would be a tedious exercise (note how complicated the recurrence relation is), so the final answer is simply left in this form.