Online Math Tutor LLC

Discover the Path to Mathematical Mastery 
An Education That Counts

Online Math Tutor LLC

Discover the Path to Mathematical Mastery 
An Education That Counts

Practice Problems: Substitution

Substitution methods are processes for transforming a given equation to reveal a particular structure. Substitution is a powerful technique for solving differential equations, particularly for those that cannot be solved using other methods. Bernoulli differential equations, which are a type of nonlinear first-order differential equation, can often be solved using substitution. This technique involves introducing a new variable and function that relates the original variables in a specific way. In this tutorial, we will explore the steps involved in solving differential equations by substitution, with a focus on Bernoulli differential equations.

Some differential equations may require a substation before we can determine how to solve them.

Homogenous EquationsBernoulli’ Differential EqutionsReduction to
Separation of Variables
\(M(x,y)dx+N(x,y)dy=0\)\(\displaystyle\frac{dy}{dx}+P(x)y=f(x)y^n\)\(\displaystyle\frac{dy}{dx}=f(Ax+By+C)\)

How do we know when a substitution is a good substitution?

Practice Problems

Determine the function that satisfies the given differential equation. 

Solve the given differential equation by making a strategic substitution.  

  1. \(\displaystyle (x+ye^{\frac{y}{x}})dx – xe^{\frac{y}{x}}dy=0\)
  2. \(\displaystyle ydy – (x+\sqrt{xy})dy=0\)
  1. \(\displaystyle x\frac{dy}{dx}=y+\sqrt{x^2-y^2}\; (x>0)\)
  2. \(\displaystyle y’=\texttt{cos}(x+y)\)
  1. \(\displaystyle y’=(x+y+1)^2\)
  2. \(\displaystyle \frac{dy}{dx}=\frac{y-x}{x+y}\)

Solve the given Bernoulli differential equation.

  1. \(\displaystyle xy’+y=y^{-2}\)
  2. \(\displaystyle xy’-(1+x)y=xy^2\)
  1. \(\displaystyle x\frac{dy}{dx}+y=2xy^3\)
  2. \(\displaystyle \frac{dy}{dx}+\frac{y}{x}=x\texttt{ln}(x)y^2\)
  1. \(\displaystyle \frac{dy}{dx}+\frac{y}{x}=3y^2 \)
  2. \(\displaystyle x^2y’+y^2=xy\)

Virtual Lessons

Need some additional help understanding how to apply this technique? Click Here to visit the virtual lesson section.