Differential Equations Decoded 7 Separable Equations Your First Superpower

Free to reuse. Free to remix. No attribution required. Make your own at https://www.patreon.com/cw/MadSciHub QUICK SUMMARY Separable equations are the first differential equations you can actually solve. Treat dy over dx as a fraction, get every y on one side and every x on the other, integrate both sides, and out comes the answer. Master this and you unlock exponential growth, cooling, radioactive decay, and half the word problems in the course. KEY CONCEPTS 1. The Divorce Move - A separable equation is one where you can force all the y stuff to one room and all the x stuff to the other room. 2. Leibniz Notation Is a Cheat Code - Treating dy over dx like a fraction is not rigorous but it works and your professor will use it. 3. One C, Not Two - Both integrals produce a constant, but you combine them into a single C on the right side. 4. Plus Or Minus Stays Until IVP - The absolute value from the natural log gives a plus or minus that only collapses when you get initial conditions. 5. Implicit Is Often Final - Sometimes y cannot be solved for explicitly and leaving the answer implicit is the correct final form. DEFINITIONS - Separable ODE: A first-order ODE that can be written as dy over dx equals f of x times g of y. - General Solution: The family of solutions containing the arbitrary constant C. - Particular Solution: The one solution picked out by an initial condition. - Initial Value Problem: A differential equation plus a starting point y of x zero equals y zero. - Equilibrium Solution: A constant solution y equals c where g of c equals zero, often lost when you divide by g of y. - Implicit Solution: A solution left as an equation in x and y that you cannot or need not solve for y. HOW IT WORKS 1. Recognize the form. Can you write the right side as a function of x times a function of y. 2. Separate. Move every y and dy to the left, every x and dx to the right. 3. Integrate both sides. Each side gets its own antiderivative. 4. Combine constants. Drop the left C, keep one C on the right. 5. Apply initial conditions if you have them. This kills C and resolves the plus or minus. 6. Solve for y if possible. If not, leave the answer implicit. 7. Check for lost solutions. If you divided by g of y, test whether g of y equals zero gives a real equilibrium you just threw away. KEY ARGUMENTS 1. Leibniz wrote this trick down in sixteen ninety one. It works because the chain rule says it works when you unwind it. 2. Treating dy over dx as a fraction is not a sin here. It is the exact operation the chain rule licenses. 3. The hardest step is almost never the calculus. It is the algebra of rearranging so variables separate cleanly. 4. The constant C is one constant, not two. The difference of two arbitrary constants is still one arbitrary constant. 5. Absolute value bars from the natural log integral produce a plus or minus that only collapses under initial conditions. 6. An implicit final answer is a feature, not a failure. If y cannot be isolated, the equation itself is the solution. 7. Dividing by g of y can silently erase an equilibrium solution. Always check what happens when g of y equals zero. 8. Exponential growth, Newton cooling, and radioactive decay are all the same separable equation in costume. KEY TAKEAWAYS - Separable means the right side factors into a function of x times a function of y with no mixing. - The move is mechanical. Separate, integrate, add one C, solve or leave implicit, apply initial conditions last. - Leibniz notation is not rigorous but the fraction trick is backed by the chain rule and it always works. - Lost equilibrium solutions are the most common hidden error. Check g of y equals zero before you submit. - If you can solve a separable equation you can model exponential growth, cooling, decay, and most first week word problems. MEMORY HOOKS - Separable means a clean divorce. Every x goes to the x apartment, every y goes to the y apartment, no custody disputes. - dy over dx looks like a fraction, walks like a fraction, and for separable equations it is a fraction. - One equation, one C. Two integrals make two constants, but you merge them into one and move on. - Plus or minus is the ghost of the absolute value. It only leaves when you feed it an initial condition. - Implicit answers are legal tender. If y will not come out alone, ship the equation and take the points. SOURCE https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/pages/syllabus/ #DifferentialEquations #SeparableEquations #Calculus #MathStudyGuide #ExamPrep #Decoded #MIT #STEM #CollegeMath #Leibniz #madscilecture #decoded #differentialequations #math #science