--- title: "What a Differential Equation Says and What a Solution Is" description: "A differential equation involves an unknown function and its derivatives. Check solutions on an interval. General solutions may contain arbitrary constants; initial data can determine a particular sol" canonical: https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o source: https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # What a Differential Equation Says and What a Solution Is A differential equation involves an unknown function and its derivatives. Check solutions on an interval. General solutions may contain arbitrary constants; initial data can determine a particular solution. Subject: Mathematics · Area: Differential Equations · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o ## Ready when they can - Verify that a proposed function satisfies a given equation - State an equation's order and count independent arbitrary constants in the regular linear solution families studied - Use an initial condition to fix the constants in a general solution ## Lesson: Equations about change, functions as answers A differential equation involves an unknown function and one or more of its derivatives. A solution has the required derivatives and satisfies the equation throughout an interval. Check it by differentiating and substituting. The derivative of e^(kx) is k e^(kx). For example, y=2e^(3x) gives y'=6e^(3x)=3y, so it solves y'=3y for every real x. Order is the highest derivative present. The symbols y', y'' and y''' mean first, second and third derivatives. A power such as (y')² still uses only the first derivative. In a linear equation, y and its derivatives are not multiplied together or raised to powers. With continuous coefficients and forcing term, and a highest-derivative coefficient that stays nonzero, an order-n general solution has n independent constants on the interval. Call this a regular linear equation. Initial conditions specify values at a chosen input. Substitute them into the solution family to determine its constants. In y=C e^x, y(0)=7 gives C=7 because e^0=1. For y=A cos x+B sin x, y'=-A sin x+B cos x. Since cos 0=1 and sin 0=0, y(0)=6 gives A=6, while y'(0)=-2 gives B=-2. *(drawing: The function value fixes A. The derivative value fixes B.)* Integrate y'''=6: y''=6x+K1, y'=3x²+K1x+K2, and y=x³+(K1/2)x²+K2x+K3. Each integration introduces an independent constant. For y''=8, y=4x²+Ax+B; 4 is fixed, while A and B vary. Differentiate twice to get y''=8. For y''=0, y=Px+Q. A condition such as y(0)=9 fixes Q but leaves P free. A residual is the equation's left side minus its right side after substitution. For y'=y, it is y'-y. A nonzero residual at one point disproves a proposed solution on any interval containing that point. A zero residual at one point is not enough: verification requires zero throughout the interval. To check an initial-value problem, also test each initial condition separately. **Recap.** Check derivatives and initial data separately. Order is the highest derivative present. Regular linear order-n families have n independent constants. One initial value may leave constants free. *Lesson, practice questions and flashcards written for this dataset.* ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk) - [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW) ## Opens up - [Steady-State and Pre-Equilibrium Approximations](https://lightmysky.com/learn/science/steady-state-and-pre-equilibrium-approximations-mt_9ESWVl28-3) - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR) - [Damped Oscillations and the Three Damping Regimes](https://lightmysky.com/learn/science/damped-oscillations-and-the-three-damping-regimes-mt_JCpYpFVn6s) - [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T)