Solving First Order Ordinary Differential Equations

Separable equation

When $x$ and $y$ functions can be separated into separate one-variable functions (as shown below).

$$\frac{\text{d}y}{\text{d}x} = f(x)g(y)$$ $$\int{\frac{1}{g(y)}\,\text{d}y} = \int{f(x)\,\text{d}x}$$

Homogenous equation

$$\frac{\text{d}y}{\text{d}x} = f(x,y)$$

A function $f(x,y)$ is homogenous when $f(x,y) = f(\lambda x, \lambda y)$.

To solve:

• Use $y = vx$ substitution, where $v$ is a function of $x$ and $y$
• Differentiate both sides: $\text{d}y = v + v\text{d}x$
• Apply the substitution to make it separable

Reduction to homogenous type

$$\frac{\text{d}y}{\text{d}x} = \frac{ax + by + c}{Ax + By + C}$$

This type of equation can be reduced to homogenous form.

If $a:b=A:B$, use the substitution: $u=ax+by$.

In other cases:

• Find $h$ and $k$ such that $ah + bk + c = 0$ and $Ah + Bk + C = 0$
• Use substitutions:
  • $X = x + h$
  • $Y = y + k$

The reduced equation would be:

$$\frac{\text{d}Y}{\text{d}X} = \frac{aX + bY}{AX + BY}$$

Linear equation

$$\frac{\text{d}y}{\text{d}x} + P(x)\,y = Q(x)$$

The above form is called the standard form.

The equation would be separable if $Q(x) = 0$. $$

Otherwise:

• Identify $P(x)$ from the standard form
• Calculate integrating factor: $I = \exp{\int{P(x) \text{d}x}}$.
• Multiply both sides by $I$
• $\text{LHS}$ becomes $\frac{d}{dx}(yI)$
• Integrate both sides to solve for $y$

Bernoulli’s equation

$$\frac{\text{d}y}{\text{d}x} + P(x)y = Q(x)y^n \;\; ; \; n\in\mathbb{R}$$

When $n = 0$ or $n = 1$, the equation would be linear.

Otherwise, it can be converted to linear using $v = y^{1-n}$ as substituion. $$

None of the above

The equation must be converted to one of the above by using a suitable substitution.