Series from Differential Equations

Use the differential equation and initial value to find successive derivatives at zero, then substitute them into the Maclaurin formula. Differentiate the whole equation, using product and chain rules where needed.

Example

For \(y^{\prime}=x+y\) and \(y(0)=1\):

\(y^{\prime}(0)=1,\qquad y^{\prime\prime}=1+y^{\prime},\qquad y^{\prime\prime}(0)=2\).

\(y^{\prime\prime\prime}=y^{\prime\prime},\qquad y^{\prime\prime\prime}(0)=2\).

\(y=1+x+x^2+\dfrac{x^3}{3}+\cdots\).

Alternatively put \(y=a_0+a_1x+a_2x^2+\cdots\) into the equation and equate coefficients. The initial value determines \(a_0\); the equation determines subsequent coefficients.