A Equation in which all x and dx can be isolated from all y and all dy.

Steps to Solve a Separable Differential Equation

<aside> 💡 Remember the C that is resulted from integration. Also that the value of C vary with the manipulation process, since is is different C value that use one unifying symbol.

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$\frac {dy} {dx}= \frac x y$

1. Manipulate and isolate all x and dx away from all y and all dy. $y dy= x dx$
2. Integrate both sides (remember the Cs.) $\int {ydy}=\int {xdx}⇒ \frac 1 2 y^2+C_1=\frac 1 2 x^2 +C_2$
   1. move the Cs to form a new C. $\frac 1 2 y^2+C_1=\frac 1 2 x^2 +C_2 ⇒\frac 1 2 y^2=\frac 1 2 x^2 +C_3$
3. Manipulate and isolate y on one side. →{y=} function. $\frac 1 2 y^2=\frac 1 2 x^2 +C_3 ⇒y^2=x^2+C_4⇒y=\sqrt {x^2+C_4}$
4. (If it is asking for a particular solution) e.g. (0, 1)
   1. pug in the points given into the {y=} function $1=\sqrt {0^2+C_4}$
      1. (or the former step(s) that have a consistent C value as the final equation.) $1=0+C_4$
      2. (or solve the particular C and led it to the final C) $\frac 1 2 =0+C_3$ → Since $C_3\cdot2=C_4$, therefore…
   2. Solve for C $1=\sqrt {0^2+C_4}⇒1=C_4$
   3. pug C into the {y=} function. $y=\sqrt {x^2+1}$