Linear equation
\[\frac{dy}{dx}+Py=Q:\quad \text{IF}=e^{\int P\,dx},\quad y\,\text{IF}=\int Q\,\text{IF}\,dx+C\]
Differential Equations
Linear Differential Equations
Understand
A first-order linear equation has the form \(\frac{dy}{dx}+P(x)\,y=Q(x)\). Multiply by the integrating factor \(\text{IF}=e^{\int P\,dx}\); the left side becomes \(\frac{d}{dx}(y\cdot\text{IF})\), so
\[y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C.\]
Example: \(\frac{dy}{dx}+\frac yx=x^2\) (\(x>0\)). \(\text{IF}=e^{\int\frac1x dx}=x\). Then \(xy=\int x^3dx=\frac{x^4}4+C\).
Similarly \(\frac{dx}{dy}+P_1(y)\,x=Q_1(y)\) is linear in \(x\): \(\text{IF}=e^{\int P_1dy}\) and \(x\cdot\text{IF}=\int Q_1\cdot\text{IF}\,dy+C\).
Key Concepts
- Write in standard form first (coefficient of dy/dx must be 1).
- IF = e^{∫P dx}; simplify e^{log u} = u.
- y·IF = ∫Q·IF dx + C.
Formula Bank
Common Mistakes
Standard form first
For \(x\frac{dy}{dx}+2y=x^2\), divide by \(x\) first: \(P=\frac2x\), \(\text{IF}=x^2\) – not \(e^{\int2\,dx}\).
Solved Examples
Linear equation
Solve \(\frac{dy}{dx}+y=e^{x}\).
\(\text{IF}=e^x\). \(ye^x=\int e^{2x}dx=\frac{e^{2x}}2+C\Rightarrow y=\frac{e^x}2+Ce^{-x}\).
Practice & Topic Test
Topic Practice
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Start PracticeTopic Summary
Standard form → integrating factor → integrate Q·IF → divide by IF if y is wanted explicitly.
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