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First-order linear equations

Differential equations

Japanese school year: University year 2

What you learn

Learn techniques for solving first-order linear differential equations, including integrating factors and variation of parameters. These equations are widely used to analyze electric circuits and mixing processes in engineering and physics. Prior mastery of the product rule for differentiation and standard integration techniques is essential.

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Key points

This is the standard form of a first-order linear differential equation, where unknown function yy and its derivative y′y' appear linearly. Here P(x)P(x) and Q(x)Q(x) are given functions of xx.

y′+P(x)y=Q(x)y' + P(x)y = Q(x)

This formula gives the general solution yy of a first-order linear differential equation directly. It is derived using the integrating factor e∫P(x) dxe^{\int P(x)\,dx}, where CC is an arbitrary constant.

y=e−∫P(x) dx(∫Q(x)e∫P(x) dx dx+C)y = e^{-\int P(x)\,dx}\left(\int Q(x)e^{\int P(x)\,dx}\,dx + C\right)

This gives the solution to the homogeneous linear equation where the right-hand side is 0. It is solved easily by separating variables, yielding an exponential function scaled by constant CC.

y′+P(x)y=0  ⟹  y=Ce−∫P(x) dxy' + P(x)y = 0 \implies y = Ce^{-\int P(x)\,dx}

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