HS Math Solutions - Calculus: Differential Equations

Calculus: Differential Equations

Contents (2 topics • 18 videos)

Separable Differential Equations (12 videos)

First-Order Linear Differential Equations (6 videos)

Separable Differential Equations

Introduction Videos

Separable Differential Equations with General Solutions

Separable Differential Equations with Particular Solutions

Theorems and Definitions

Separable Differential Equations The first-order differential equation of the form \[ \frac{dy}{dx} = H(x, y) \] is called separable if the variables \(x\) and \(y\) can be separated and the equation can be written in the form \[ f(y) \text{ } dy = g(x) \text{ } dx \]

Video Practice Problems Playlist on YouTube

Find the general solution.
\[ \frac{dy}{dx}=\frac{12x^3}{4y-\sin{y}} \]

Find the general solution.
\[ \frac{dy}{dx}=3x \sqrt{y} \]

Find the general solution.
\[ x y'=3(y-2) \]

Find the general solution.
\[ \frac{dy}{dx}=e^{x-2y} \]

Use the initial condition to find the particular solution.
\[ y \cdot y'-5e^x=10 \quad \quad y(0)=2 \]

Use the initial condition to find the particular solution.
\[ 2y \cdot y'=4 \sin{x} \quad \quad y \Big( \frac{\pi}{4} \Big) =\sqrt{2} \]

Use the initial condition to find the particular solution.
\[ \sqrt{x} -\sqrt{y} \cdot y' = 0 \quad \quad y(9)=1 \]

Use the initial condition to find the particular solution.
\[ y(2x-1)+y'=0 \quad \quad y(-3)=e \]

Use the initial condition to find the particular solution.
\[ y \cdot \ln{x} - xy'=0 \quad \quad y \big(e^2 \big)=1 \]

Use the initial condition to find the particular solution.
\[ y' = x y \sin{x^2} \quad \quad y(0)=\sqrt{e} \]

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

First-Order Linear Differential Equations

Introduction Videos

First-Order Linear Differential Equations

Theorems and Definitions

First-Order Linear Differential Equations A first-order linear differential equation is an equation of the form \[ \frac{dy}{dx} + P(x) \cdot y=Q(x) \] where \(P\) and \(Q\) are continuous functions of \(x\). This first-order linear differential equation is said to be in standard form.

An integrating factor for the first-order linear differential equation is \[ r(x) = e^{ \int{ P(x) \text{ } dx }}. \] The solution of the differential equation is \[ y \cdot r(x) = \int{ Q(x) \cdot r(x) \text{ } dx }. \]

Video Practice Problems Playlist on YouTube

Find the general solution.
\[ y'+y=3 \]

Find the general solution.
\[ y'+y=e^{-x} \]

Find the general solution.
\[ 2xy'+y=10\sqrt{x} \]

Use the initial condition to find the particular solution.
\[ xy'-4y=6x^{3} \quad \quad y(2)=24 \]

Use the initial condition to find the particular solution.
\[ xy'+3y=\frac{\ln x}{x^{2}} \quad \quad y(1)=3 \]

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)