Differential Equations

To determine the constant of integration.

To simplify the equation and solve for one variable at a time.

To apply initial conditions and ensure consistency.

To integrate both sides of the equation independently.

Differentiate immediately again to ensure that the integrals have been properly calculated.

Join the integrated function back together into a differential equation using the chain rule.

Nestily exponentiate both sides if the integral resulted in an exponential relationship or to eliminate the logarithmic term.

Apply boundary conditions immediately to determine the arbitrary constant value.

The value of $d$ must be greater than zero to ensure unbounded growth for $x$ as $t$ becomes large.

Decreasing $d$ below one will cause $x$ to grow without bound as $t$ exceeds one.

Increasing $d$ above one ensures exponential growth for all $t$ greater than one.

The value of $d$ does not affect unbounded growth since only values of $t$ determine this behavior.

y(x) = \ln(x)

y(x) = \ln(x) - 1

y(x) = \ln(x) + 1

y(x) = \ln(x) + 2

y(x) = 2\cos(x)

y(x) = \sin(x) + C

y(x) = 2\sin(x) + C

y(x) = \cos(x)

Explicit functions that describe the behavior of the system.

Sets or families of functions that satisfy the equation conditions.

Differential equations with infinite solutions.

Specific solutions that depend on a constant value.

Differentiate both sides with respect to $x$.

Multiply both sides by $dx$ and divide by $y$.

Integrate both sides as they are.

Take the natural logarithm of both sides.

For all x excluding any points where h(y)=0 to avoid division by zero

Continuity of g(x) is not required for separation of variables to work effectively

Only at the initial condition provided by an initial value problem

For all x in the domain where separation occurs

$y = Ce^{-\sin(x)}$

$y = Ce^{\cos(x)}$

$y = Ce^{\sin(x)}$

$y = Ce^{-\cos(x)}$

y(x) = 2x

y(x) = x^2 + 3

y(x) = 2x + 3

y(x) = x^2