7.2 Separable Differential Equations
The equation $y^{\prime}=f(x, y)$ is a separable equation if all $x$ terms can be collected with $d x$ and all $y$ terms with $d y$. The differential equation then has the form
$
\frac{d y}{d x}=f(x) g(y) \text { or } \frac{d y}{d x}=\frac{f(x)}{h(y)} \text {. }
$
To solve the first equation we could rewrite it in the form $\frac{d y}{g(y)}=f(x) d x$, and integrate both sides of the equation:
$
\int \frac{d y}{g(y)}=\int f(x) d x
$
To solve the second equation we could rewrite it in the form $h(y) d y=f(x) d x$ and integrate both sides of the equation:
$
\int h(y) d y=\int f(x) d x
$
Example 3
- Find the general solution of $(x+3) y^{\prime}=2 y$.
▶️Answer/Explanation
Solution
$
\begin{aligned}
& (x+3) y^{\prime}=2 y \\
& (x+3) \frac{d y}{d x}=2 y\quad\quad\quad\text{Rewrite $y^{\prime}$ as $\frac{d y}{d x}$} \\
& \frac{d y}{y}=\frac{2}{x+3} d x\quad\quad\quad\text{Separate the variables} \\
& \int \frac{d y}{y}=\int \frac{2}{x+3} d x\quad\quad\quad\text{Integrate} \\
& \ln |y|=2 \ln |x+3|+C_1 \\
& =\ln (x+3)^2+\ln C\quad\quad\text{Let $C_1=\ln C$.} \\
& y=C(x+3)^2\quad\quad\quad\quad\text{General solution}
\end{aligned}
$
Example 4
$
\text { Find the general solution of } \frac{d y}{d x}=-\frac{2 x}{y} \text {. }
$
▶️Answer/Explanation
Solution
$
\begin{array}{ll}
\frac{d y}{d x}=-\frac{2 x}{y} & \\
y d y=-2 x d x & \text { Separate the variables. } \\
\int y d y=\int-2 x d x & \text { Integrate. } \\
\frac{1}{2} y^2=-x^2+C_1 & \\
2 x^2+y^2=C & \text { General solution, } C=2 C_1
\end{array}
$
Example 5
- The solution to the differential equation $\frac{d y}{d x}=\frac{3 x^2}{2 y}$, where $y(3)=4$, is
(A) $y=\sqrt{\frac{x^3}{3}}+1$
(B) $y=7-\sqrt{\frac{x^3}{3}}$
(C) $y=\sqrt{x^3-9}$
(D) $y=\sqrt{x^3-11}$
▶️Answer/Explanation
Ans:D
Example 6
- If $\frac{d y}{d x}=\frac{x+\sec ^2 x}{y}$ and $y(0)=2$, then $y=$
(A) $\sqrt{x^2+2 \sec x+2}$
(B) $\sqrt{x^2+2 \tan x+4}$
(C) $\sqrt{x^2+\sec ^2 x+2}$
(D) $\sqrt{x^2+\tan ^2 x+4}$
▶️Answer/Explanation
Ans:B
Example 7
At each point $(x, y)$ on a certain curve, the slope of the curve is $x y$. If the curve contains the point $(0,-1)$, which of the following is the equation for the curve?
(A) $y=x^2-2$
(B) $y=3 x^2-4$
(C) $y=-e^{\frac{x^2}{2}}$
(D) $y=-e^{\left(x^2-1\right)}$
▶️Answer/Explanation
Ans:C