Question

Most differential equations used in science cannot be solved analytically. That is, there is no way to write a solution in terms of elementary functions. One approach to these problems is Euler's method, which provides an approximate numerical solutions. Here we'll consider another approach which expresses the solution as a power series. Recall that the differential equation y' = ky has the solution y(x) = Aekx. As an alternate derivation, we'll write the solution y(x) as a power series centered at x = 0. y(x) = sum_{n=0}^{infty} a_nx^n = a_0 + a_1x + a_2x^2 + ... Substituting this into the differential equation we get frac{d}{dx}(a_0 + a_1x + a_2x^2 + ...) = k(a_0 + a_1x + a_2x^2 + ...) a_1 + 2a_2x + 3a_3x^2 + ... = ka_0 + ka_1x + ka_2x^2 + ... For this equation to hold, the coefficients of each power of x must be equal. This means that a_1 = ka_0 2a_2 = ka_1 3a_3 = ka_2 : Using this sequence of equations we can find a general expression for the coefficients. a_1 = ka_0 a_2 = frac{ka_1}{2} = frac{k^2a_0}{2} a_3 = frac{ka_2}{3} = frac{k^3a_0}{3 cdot 2} : a_n = frac{k^na_0}{n!} So the solution is y(x) = a_0 + ka_0x + frac{k^2a_0}{2}x^2 + frac{k^3a_0}{2 cdot 3}x^3 ... = a_0sum_{n=0}^{infty} frac{(kx)^n}{n!} = a_0e^{kx} which agrees with our previous result.

          Most differential equations used in science cannot be solved analytically. That is, there is no way to write a solution in terms of elementary functions. One approach to these problems is Euler's method, which provides an approximate numerical solutions. Here we'll consider another approach which expresses the solution as a power series.
Recall that the differential equation
y' = ky
has the solution y(x) = Aekx. As an alternate derivation, we'll write the solution y(x) as a power series centered at x = 0.
y(x) = sum_{n=0}^{infty} a_nx^n = a_0 + a_1x + a_2x^2 + ...
Substituting this into the differential equation we get
frac{d}{dx}(a_0 + a_1x + a_2x^2 + ...) = k(a_0 + a_1x + a_2x^2 + ...)
a_1 + 2a_2x + 3a_3x^2 + ... = ka_0 + ka_1x + ka_2x^2 + ...
For this equation to hold, the coefficients of each power of x must be equal. This means that
a_1 = ka_0
2a_2 = ka_1
3a_3 = ka_2
:
Using this sequence of equations we can find a general expression for the coefficients.
a_1 = ka_0
a_2 = frac{ka_1}{2} = frac{k^2a_0}{2}
a_3 = frac{ka_2}{3} = frac{k^3a_0}{3 cdot 2}
:
a_n = frac{k^na_0}{n!}
So the solution is
y(x) = a_0 + ka_0x + frac{k^2a_0}{2}x^2 + frac{k^3a_0}{2 cdot 3}x^3 ... = a_0sum_{n=0}^{infty} frac{(kx)^n}{n!} = a_0e^{kx}
which agrees with our previous result.

$Most differential equations used in science cannot be solved analytically. That is, there is no way to write a solution in terms of elementary functions. One approach to these problems is Euler's method, which provides an approximate numerical solutions. Here we'll consider another approach which expresses the solution as a power series. Recall that the differential equation y' = ky has the solution y(x) = Aekx. As an alternate derivation, we'll write the solution y(x) as a power series centered at x = 0. y(x) = sumn=0^infty anx^n = a0 + a1x + a2x^2 + ... Substituting this into the differential equation we get fracddx(a0 + a1x + a2x^2 + ...) = k(a0 + a1x + a2x^2 + ...) a1 + 2a2x + 3a3x^2 + ... = ka0 + ka1x + ka2x^2 + ... For this equation to hold, the coefficients of each power of x must be equal. This means that a1 = ka0 2a2 = ka1 3a3 = ka2 : Using this sequence of equations we can find a general expression for the coefficients. a1 = ka0 a2 = fracka12 = frack^2a02 a3 = fracka23 = frack^3a03 cdot 2 : an = frack^na0n! So the solution is y(x) = a0 + ka0x + frack^2a02x^2 + frack^3a02 cdot 3x^3 ... = a0sumn=0^infty frac(kx)^nn! = a0e^kx which agrees with our previous result.$

Added by Marvin W.

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