Solution to Separable Differential Equation/Examples/Arbitrary Example 1

Example of Use of Solution to Separable Differential Equation

Consider the first order ODE:

$(1): \quad \map {\dfrac \d {\d x} } {\map f x} = 3 x$

where we are given that $\map f 1 = 2$.

The particular solution to $(1)$ is:

$\map f x = \dfrac {3 x^2 + 1} 2$

Proof

We use Solution to Separable Differential Equation:

\(\ds \map {\dfrac \d {\d x} } {\map f x}\)

\(=\)

\(\ds 3 x\)

\(\ds \leadsto \ \ \)

\(\ds \map f x\)

\(=\)

\(\ds \int 3 x \rd x\)

\(\ds \)

\(=\)

\(\ds \dfrac {3 x^2} 2 + C\)

When $x = 1$, we have that $\map f x = 2$.

Hence:

\(\ds \dfrac {3 \times 1^2} 2 + C\)

\(=\)

\(\ds 2\)

\(\ds \leadsto \ \ \)

\(\ds C\)

\(=\)

\(\ds 2 - \dfrac 3 2\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 2\)

Hence:

$\map f x = \dfrac {3 x^2 + 1} 2$

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Exercises: $6.1$ The Primitive Function: $1. \ \text {(a)}$