Differential Equation Solution and Types of Differential Equations
Step-by-step explanation of solving a first-order linear differential equation and an overview of common differential equation types.
Solving the Differential Equation
Given differential equation:
$$\frac{dx}{dt} = 10 - 0.2x$$
1. Rearrange the equation
Move the \(x\) term to the left:
$$\frac{dx}{dt} + 0.2x = 10$$
This is a first-order linear differential equation of the form:
$$\frac{dx}{dt} + P(t)x = Q(t)$$
Here:
- P(t) = 0.2
- Q(t) = 10
2. Find the Integrating Factor
$$IF = e^{\int P(t)dt}$$
$$IF = e^{\int 0.2dt}$$
$$IF = e^{0.2t}$$
3. Multiply the equation by the integrating factor
$$e^{0.2t}\frac{dx}{dt} + 0.2e^{0.2t}x = 10e^{0.2t}$$
The left side becomes:
$$\frac{d}{dt}(xe^{0.2t}) = 10e^{0.2t}$$
4. Integrate both sides
$$\int \frac{d}{dt}(xe^{0.2t})dt = \int 10e^{0.2t}dt$$
Result:
$$xe^{0.2t} = 50e^{0.2t} + C$$
5. Solve for x
Divide by \(e^{0.2t}\):
$$x(t) = 50 + Ce^{-0.2t}$$
Final General Solution
$$x(t) = 50 + Ce^{-0.2t}$$
Interpretation: The value of \(x(t)\) approaches 50 over time because the term \(Ce^{-0.2t}\) decays to zero.
Types of Differential Equation Cases
Yes — there can be several other types of differential equation cases, including the one shown earlier (which is a first-order linear differential equation solved using an integrating factor).
Your example:
$$\frac{dx}{dt}=10-0.2x$$
Rewritten as:
$$\frac{dx}{dt}+0.2x=10$$
This is a first-order linear ODE.
1. Separable Differential Equations
If the equation can be written as:
$$\frac{dx}{dt}=f(t)g(x)$$
then we separate variables:
$$\frac{dx}{g(x)} = f(t)dt$$
Example:
$$\frac{dx}{dt}=tx$$
Separate variables:
$$\frac{1}{x}dx = tdt$$
Integrate:
$$\ln|x|=\frac{t^2}{2}+C$$
Solution:
$$x = Ce^{t^2/2}$$
2. Linear First-Order Differential Equation
This is exactly the case used earlier.
General form:
$$\frac{dx}{dt}+P(t)x=Q(t)$$
Solved using an integrating factor:
$$IF=e^{\int P(t)dt}$$
Example:
$$\frac{dx}{dt}+3x=6$$
3. Homogeneous Differential Equation
If the equation has the form:
$$\frac{dy}{dx}=F\left(\frac{y}{x}\right)$$
Substitute:
$$v=\frac{y}{x}$$
Example:
$$\frac{dy}{dx}=\frac{x+y}{x}$$
4. Exact Differential Equations
Form:
$$M(x,y)dx+N(x,y)dy=0$$
If
$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$
then the equation is exact.
Example:
$$ (2xy+3)dx + (x^2+4y)dy = 0 $$
5. Bernoulli Differential Equation
Form:
$$\frac{dy}{dx}+P(x)y=Q(x)y^n$$
Example:
$$\frac{dy}{dx}+y=xy^2$$
Solve using substitution:
$$v=y^{1-n}$$
6. Logistic Growth Equation
This is common in biology and population models.
Form:
$$\frac{dx}{dt}=rx\left(1-\frac{x}{K}\right)$$
Solution:
$$x(t)=\frac{K}{1+Ce^{-rt}}$$
7. Second-Order Differential Equations
Example:
$$\frac{d^2x}{dt^2}+3\frac{dx}{dt}+2x=0$$
Solve using the characteristic equation.
Your Equation Type
Your equation:
$$\frac{dx}{dt}=10-0.2x$$
This is also called a linear decay toward equilibrium model where:
- Equilibrium value = 50
- Rate constant = 0.2
Solution:
$$x(t)=50+Ce^{-0.2t}$$
Summary
Besides the linear first-order case, there are many other differential equation types:
- Separable equations
- Homogeneous equations
- Exact equations
- Bernoulli equations
- Logistic equations
- Higher-order differential equations