Can you show a worked example?

Solve dy/dx = 2x with y(0) = 3. y = x² + 3.

Separable differential equations

Separate, integrate both sides, then use the initial condition to find C.

The common mistake

Separating and integrating correctly, then stopping with a general solution and never using the initial condition to find C.

Instead: Put all y terms with dy and all x terms with dx, integrate both sides, add one + C, then substitute the given point to solve for C before writing y =.

Worked example with Guide me

Solve dy/dx = 2x with y(0) = 3.

Sugar · Separate. What do you get?
Student · dy = 2x dx.
Sugar · Integrate both sides. What's y?
Student · y = x² + C.
Sugar · Use y(0) = 3. What's C?
Student · 3 = 0 + C, so C = 3.

Answer: y = x² + 3.

Try 3 practice questions

1. Solve dy/dx = 6x² with y(0) = 1.
Your answer
2. Solve dy/dx = 4 with y(1) = 10.
Your answer
3. Solve dy/dx = y with y(0) = 5.
Your answer

Quick answers

What is the most common mistake with separable differential equations?: Separating and integrating correctly, then stopping with a general solution and never using the initial condition to find C.
How do I get separable differential equations right?: Put all y terms with dy and all x terms with dx, integrate both sides, add one + C, then substitute the given point to solve for C before writing y =.
Can you show a worked example?: Solve dy/dx = 2x with y(0) = 3. y = x² + 3.

Related concepts

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