Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain what a general solution is and why it contains one arbitrary constant. (Guide: “From checking solutions to finding them”; Figure 1)
- I can explain why one constant is enough when I integrate both sides. (Guide: “One constant is enough”)
- I can explain, using the chain rule, why separating and integrating both sides gives a correct solution. (Guide: “Why this is valid”)
- I can decide whether a differential equation is separable, factoring or using exponent laws first where needed. (Guide: separable table; Practice Q1, Q6(c))
Skills
- I can find the general solution of dy/dx = f(x) by antidifferentiating. (Guide: “The simplest case”)
- I can separate the variables and antidifferentiate both sides with the correct antiderivatives. (Guide: Worked example 1; Practice Q2, Q5)
- I can tell when a fraction gives a logarithm and when it does not, for example ∫ (1/y²) dy = −1/y. (Guide: “Not every fraction gives a logarithm”; Practice Q2, Q3)
- I can turn ln|y + b| = G(x) + C into y = A e^(G(x)) − b, with the constant as a multiplier. (Guide: Worked example 2; Practice Q4, Q6(c))
- I can split e^(x + y) or e^(x − y) into a product and solve. (Guide: Worked example 3)
- I can leave a solution in implicit form when y cannot be isolated. (Guide: “When you cannot solve for y”; Practice Q7)
Reasoning
- I can check a general solution by differentiating, explicitly or implicitly. (Guide: Worked examples 1 to 3; Practice Q5(c), Q7(b))
- I can explain why y = e^(G(x)) + C and y = tan(x²) + C are wrong forms. (Guide: “Watch the constant”; Practice Q5(c))
- I can find and explain the error when someone integrates y with respect to x without separating. (Practice Q6(a), Q6(b))
- I can say which constant solutions are lost when I divide by h(y), and how A = 0 can bring them back. (Guide: Worked example 2, step 5)
All ticked? Move on to Topic 7.7, Finding Particular Solutions Using Initial Conditions and Separation of Variables, or return to the Calculus AB roadmap or the Calculus BC roadmap.