Equations that don’t contain x or y
With differential equations that don’t explicitly contain y and differential equations that don’t explicitly contain x, the order of the equation can be reduced and solved as a lower order equation.
The Essentials
If the equation doesn’t contain y, this substitution can be used:
\[
v = \frac{dy}{dx}
\]
\[
\frac{dv}{dx} = \frac{d^2y}{dx^2}
\]
If the equation doesn’t explicitly contain x, this substitution can be used:
\[
v = \frac{dy}{dx}
\]
\[
\frac{d^2y}{dx^2} = \frac{dv}{dx} = v \frac{dv}{dy}
\]
These substitutions sometimes work on higher than second order equations.
Example
Solve this equation:
\[
y \frac{d^2y}{dx^2} = 2 \left(\frac{dy}{dx}\right)^2
\]
This equation doesn’t explicitly contain x, so we can use the substitution:
\[
yv {\frac{dv}{dy}} = 2v^2
\]
This equation is separable.
\[
\int \frac{1}{v} \, dv = \int \frac{2}{y} \, dy
\]
\[
\ln |v| = 2 \ln |y| + A
\]
\[
v = Ay^2
\]
Now that we have it solved for v, we can undo the substitution and solve for y:
\[
\frac{dy}{dx} = Ay^2
\]
\[
\int y^{-2} \, dy = \int A \, dx
\]
\[
- y^{-1} = Ax + B
\]
\[
y(x) = \frac{1}{-Ax - B}
\]
Practice
Evaluate these expressions:
-
\[ yy' y'' = 9 \]
-
\[ xy'' + y' = 4x \]
-
\[ y^{(4)} - 4y^{(3)} = 0 \]
Solution:
-
\[ \frac{1}{C_1} y + \frac{3}{C_1^2} \ln |C_1 y - 3| = x + C_2 \]
-
\[ y(x) = x^2 + C_1 \ln |x| + C_2 \]
-
\[ C_1 e^{4x} + C_2 x^2 + C_3 x + C_4 \]