Business Mathematics and Statistics · Ch 4 — Differential Equations

Linear Differential Equations of the First Order

The standard form and integrating factor

A first-order linear differential equation has the standard form

\frac{dy}{dx}+Py=Q

where $P,Q$ are functions of $x$ alone (never of $y$ ). Unlike the variable-separable case, $y$ and $x$ genuinely cannot be separated here — instead, multiplying through by the integrating factor

\text{I.F.}=e^{\int P\,dx}

makes the left-hand side collapse into the derivative of a single product, $\frac{d}{dx}(y\cdot\text{I.F.})$ . The general solution is then

y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx+C

Worked reasoning

For $\frac{dy}{dx}+y=e^x$ (so $P=1,Q=e^x$ ): $\text{I.F.}=e^{\int1\,dx}=e^x$ . The solution is

y\cdot e^x=\int e^x\cdot e^x\,dx=\int e^{2x}\,dx=\frac{e^{2x}}{2}+C

so $y=\frac{e^x}{2}+Ce^{-x}$ .

Note

The integrating factor is found from $P$ alone, never from $Q$ …

Definition 1Linear Differential Equation (First Order)

$\frac{dy}{dx}+Py=Q$ with $P,Q$ functions of $x$ alone; solved using the integrating factor $\text{I.F.}=e^{\int P\,dx}$ , giving $y\cdot\text{I.F.}=\in …

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