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# Identifying separable equations

To solve a differential equation using separation of variables, we must be able to bring it to the form f, left parenthesis, y, right parenthesis, d, y, equals, g, left parenthesis, x, right parenthesis, d, x where f, left parenthesis, y, right parenthesis is an expression that doesn't contain x and g, left parenthesis, x, right parenthesis is an expression that doesn't contain y.
Not all differential equations are like that. For example, start fraction, d, y, divided by, d, x, end fraction, equals, x, plus, y cannot be brought to the form f, left parenthesis, y, right parenthesis, d, y, equals, g, left parenthesis, x, right parenthesis, d, x no matter how much we try.
In fact, a major challenge with using separation of variables is to identify where this method is applicable. Differential equations that can be solved using separation of variables are called separable equations.
So how can we tell whether an equation is separable? The most common type are equations where start fraction, d, y, divided by, d, x, end fraction is equal to a product or a quotient of f, left parenthesis, y, right parenthesis and g, left parenthesis, x, right parenthesis.
For example, start fraction, d, y, divided by, d, x, end fraction, equals, start fraction, start color #11accd, g, left parenthesis, x, right parenthesis, end color #11accd, divided by, start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c, end fraction can turn into start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c, d, y, equals, start color #11accd, g, left parenthesis, x, right parenthesis, end color #11accd, d, x when multiplied by start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c and d, x.
Also, start fraction, d, y, divided by, d, x, end fraction, equals, start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c, start color #11accd, g, left parenthesis, x, right parenthesis, end color #11accd can turn into start fraction, 1, divided by, start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c, end fraction, d, y, equals, start color #11accd, g, left parenthesis, x, right parenthesis, end color #11accd, d, x when divided by start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c and multiplied by d, x.
Here are a few concrete examples:
\begin{aligned}\dfrac{dy}{dx}&=\overbrace{\maroonD{\sin(y)}}^{\maroonD{f(y)}}\overbrace{\blueD{\ln(x)}}^{\blueD{g(x)}}\\\\\dfrac{1}{\sin(y)}dy&=\ln(x)\,dx\\\\\end{aligned}
\begin{aligned}\dfrac{dy}{dx}&=\dfrac{\overbrace{\blueD{x^3-5x}}^{\blueD{g(x)}}}{\underbrace{\maroonD{e^y}}_{\maroonD{f(y)}}}\\\\e^y\,dy&=(x^3-5x)\,dx\\\\\end{aligned}
\begin{aligned}\dfrac{dy}{dx}&=\dfrac{\overbrace{\maroonD{\sqrt y}}^{\maroonD{f(y)}}}{\underbrace{\blueD{\cos(x)}}_{\blueD{g(x)}}}\\\\\dfrac{1}{\sqrt y}dy&=\dfrac{1}{\cos(x)}dx\end{aligned}
Other equations must be slightly manipulated before they are in the form start fraction, d, y, divided by, d, x, end fraction, equals, f, left parenthesis, y, right parenthesis, g, left parenthesis, x, right parenthesis. For example, we need to factor the right-hand side of start fraction, d, y, divided by, d, x, end fraction, equals, x, y, minus, 7, x to bring it to the desired form:
start fraction, d, y, divided by, d, x, end fraction, equals, x, y, minus, 7, x, equals, start overbrace, start color #11accd, x, end color #11accd, end overbrace, start superscript, start color #11accd, g, left parenthesis, x, right parenthesis, end color #11accd, end superscript, start overbrace, left parenthesis, start color #ca337c, y, minus, 7, end color #ca337c, right parenthesis, end overbrace, start superscript, start color #ca337c, f, left parenthesis, y, right parenthesis, end color #ca337c, end superscript
Problem 1
Can this differential equation be solved using separation of variables?
start fraction, d, y, divided by, d, x, end fraction, equals, 3, y, minus, x, squared, y

Problem 2
Can this differential equation be solved using separation of variables?
start fraction, d, y, divided by, d, x, end fraction, equals, 4, x, plus, 5, y, plus, 4