Linear Differential Equations

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Standard Form

\[y'+P(x)y=Q(x),\quad\mu=e^{{\int P\,dx}}\]

Multiply through by μ: \((\mu y)'=\mu Q\), then integrate.

Examples

Example 1. y'+y=e^x.

Solution. μ=eˣ. (eˣy)'=e^{{2x}} → y=e^x/2+Ce^{{-x}}.

In Depth

A first-order linear ODE has the form \(y' + P(x)y = Q(x)\). The integrating factor method multiplies both sides by \(\mu(x) = e^{\int P\,dx}\), making the left side the derivative of \(\mu y\): \((\mu y)' = \mu Q\). Integrating gives \(y = (1/\mu)\int \mu Q\,dx + C/\mu\).

Second-order linear ODEs with constant coefficients \(ay''+by'+cy=0\) are solved via the characteristic equation \(ar^2+br+c=0\). Three cases: (1) two distinct real roots \(r_1,r_2\): \(y=C_1e^{r_1x}+C_2e^{r_2x}\); (2) repeated root \(r\): \(y=(C_1+C_2x)e^{rx}\); (3) complex roots \(\alpha\pm\beta i\): \(y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\).

For nonhomogeneous equations \(ay''+by'+cy=f(x)\), the general solution is \(y=y_h+y_p\) where \(y_h\) is the homogeneous solution and \(y_p\) is any particular solution. Methods for finding \(y_p\): undetermined coefficients (for polynomial, exponential, sinusoidal \(f\)) and variation of parameters (general method).

Linear ODEs model spring-mass systems (\(mx''+bx'+kx=F(t)\)), RLC circuits (\(LQ''+RQ'+Q/C=V(t)\)), and beam deflection. The resonance phenomenon — when the forcing frequency matches the natural frequency — leads to unbounded oscillations in the undamped case, with critical engineering implications.

Key Properties & Applications

A first-order linear ODE \(y'+P(x)y=Q(x)\) is solved by the integrating factor \(\mu=e^{\int P\,dx}\): multiply both sides by \(\mu\) to get \((\mu y)'=\mu Q\), then integrate. This always works when \(P\) and \(Q\) are continuous.

Second-order linear ODEs \(y''+py'+qy=f(x)\) have general solution = complementary solution (solving the homogeneous equation) + particular solution. The complementary solution depends on the roots of the characteristic equation \(r^2+pr+q=0\): real distinct roots give \(e^{r_1x},e^{r_2x}\); repeated root gives \(e^{rx},xe^{rx}\); complex roots \(\alpha\pm\beta i\) give \(e^{\alpha x}\cos\beta x,e^{\alpha x}\sin\beta x\).

The method of undetermined coefficients finds particular solutions when \(f(x)\) is a polynomial, exponential, sine, or cosine (or products thereof). Variation of parameters works for any continuous \(f(x)\) and uses the Wronskian of the complementary solutions.