Particular integralsAQA A-Level Further Maths: Revision notes

For $y''+ay'+by=f(x)$ the general solution is $y=\text{CF}+\text{PI}.$ The complementary function (CF) is the general solution of the homogeneous equation $y''+ay'+by=0$ (found from the auxiliary equation) and contains two arbitrary constants. The particular integral (PI) is any one function that satisfies the full equation and contains no arbitrary constants. Adding the PI shifts the solution so that it also produces $f(x)$ on the left.

Choose the PI to match the form of $f(x)$:

• $f(x)$ a polynomial of degree $n$: a general polynomial of degree $n$, e.g. $px+q$ for $12x$.
• $f(x)=ke^{\lambda x}$: $pe^{\lambda x}$.
• $f(x)=k\cos\omega x$ or $k\sin\omega x$ (or both): $p\cos\omega x+q\sin\omega x$, always with both terms, because differentiation swaps sine and cosine. Include every lower power in a polynomial trial: $px^2+qx+r$ for a quadratic $f$.

Differentiate the trial function twice, substitute into the equation, then compare coefficients of each type of term. Example: $y''-y'-6y=12x$ with $y=px+q$ has $y'=p$ and $y''=0$. So $-p-6px-6q=12x$. Comparing $x$: $-6p=12$, so $p=-2$. Comparing constants: $-p-6q=0$, so $q=\frac13$. PI: $y=-2x+\frac13$. The CF is $Ae^{3x}+Be^{-2x}$, so the general solution is $y=Ae^{3x}+Be^{-2x}-2x+\frac13$.

If $f(x)$ is a term of the CF, the trial function gives $0$ on the left and fails. Multiply it by $x$. For $y''-3y'+2y=4e^{x}$ the CF is $Ae^{x}+Be^{2x}$, so try $y=pxe^{x}$. Then $y'=p(1+x)e^{x}$ and $y''=p(2+x)e^{x}$, and the equation gives $-pe^{x}=4e^{x}$, so $p=-4$. If a repeated root makes $e^{mx}$ and $xe^{mx}$ both part of the CF, multiply by $x^2$ instead. Always find the CF first so that you spot this.

Write the general solution first, then apply two conditions (often $y$ and $\frac{dy}{dx}$ at $x=0$) to the general solution, not to the CF alone. For $y''+3y'+2y=10\cos x$: $y=Ae^{-x}+Be^{-2x}+\cos x+3\sin x$. With $y(0)=2$ and $y'(0)=1$ we get $A+B=1$ and $A+2B=2$, so $y=e^{-2x}+\cos x+3\sin x$. For large $x$ the CF terms decay and the PI gives the steady oscillation.