Solution by Integrating Factor/Examples/y' + y = x^-1

Theorem

Consider the linear first order ODE:

$(1): \quad \dfrac {\d y} {\d x} + y = \dfrac 1 x$

with the initial condition $\tuple {1, 0}$.

This has the particular solution:

$y = \ds e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$

Proof

This is a linear first order ODE with constant coefficents in the form:

$\dfrac {\d y} {\d x} + a y = \map Q x$

where:

$a = 1$

$\map Q x = \dfrac 1 x$

with the initial condition $y = 0$ when $x = 1$.

Thus from Solution to Linear First Order ODE with Constant Coefficients with Initial Condition:

\(\ds y\)

\(=\)

\(\ds e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi + 0 \cdot e^{x - 1}\)

\(\ds \)

\(=\)

\(\ds e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi\)

From Primitive of $\dfrac {e^x} x$ has no Solution in Elementary Functions, further work on this is not trivial.

$\blacksquare$

Also see

• Primitive of Exponential of a x over x

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor: Example $2$