Chapter 6 | Substitutions | Math 2280

The subject of Chapter 6 is substitutions and how they help simplify differential equations.

Chapter 6.1 - General Substitutions

Upshot: If a differential equation 
\[
\frac{\mathrm{d} y}{\mathrm{d} x} = F(x,y) 
\]
 is not in one of the simple forms we have already seen in class (i.e. linear or separable equations), then we can try to use $u$-substitution to transform it into an easier form.

Example: If we want to start solving the equation
\[
y \frac{\mathrm{d} y}{\mathrm{d} x} + x = \sqrt{x^2 + y^2}
\]
then we see that if we make the substitution $u = x^2 + y^2$ we get
\[
\frac{\mathrm{d} u}{\mathrm{d} x} = 2x + 2y \frac{\mathrm{d} y}{\mathrm{d} x}
\]
and the substitution
\[
\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\sqrt{u}- x}{y}
\]
and thus
\[
\frac{\mathrm{d} u}{\mathrm{d} x} = 2 \sqrt{u}.
\]
We know how to solve this separable equation, thus we have successfully simplified the original differential equation.

Chapters 6.2--6.4 - Useful special substitutions

Here are the three substitutions discussed in Chapters 6.2-6.4. 


In particular in the case of linear substitutions and homogeneous substitutions we always end up with a separable differential equation and in the case of a Bernoulli equation we end up with a linear differential equation after substitution.