Number of constants
The general solution of an nth-order equation has exactly n arbitrary constants.
Differential Equations
General and Particular Solutions
Understand
A solution of a differential equation is a function that satisfies it. The general solution contains as many arbitrary constants as the order; a particular solution is obtained by giving the constants specific values, usually from an initial condition such as \(y(0)=1\).
To verify a solution, differentiate it and substitute into the equation. Example: \(y=Ae^{2x}\) satisfies \(y'=2y\) because \(y'=2Ae^{2x}=2y\).
An \(n\)th-order equation has a general solution with \(n\) independent constants; for example \(y=A\cos x+B\sin x\) is the general solution of \(y''+y=0\).
Key Concepts
- General solution: n constants for order n.
- Particular solution: constants fixed by conditions.
- Verify by substitution.
Key Points
Practice & Topic Test
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Differentiate the proposed solution and substitute; use initial conditions to find the constants.
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