Introduction to Differential Equations Math Example 2

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Example 2

medium
Find the general and particular (y(0)=4y(0)=4) solution to y′=3x2+1y' = 3x^2+1.

Solution

  1. 1
    Integrate both sides with respect to xx: dydx=3x2+1  ⟹  y=x3+x+C\frac{dy}{dx} = 3x^2+1 \implies y = x^3+x+C
  2. 2
    Apply the initial condition y(0)=4y(0) = 4 to find the constant: 03+0+C=40^3 + 0 + C = 4, so C=4C = 4.
  3. 3
    Write the particular solution: y=x3+x+4y = x^3 + x + 4

Answer

General: y=x3+x+Cy = x^3+x+C; particular: y=x3+x+4y = x^3+x+4
When the right side depends only on xx, the solution is the antiderivative. The IC determines CC.

About Introduction to Differential Equations

An equation that contains an unknown function and one or more of its derivatives. Solving a DE means finding the function(s) that satisfy the equation.

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More Introduction to Differential Equations Examples

Example 1 easy

Verify [formula] solves [formula], and find the particular solution with [formula].

Example 3 easy

Verify that [formula] satisfies [formula].

Example 4 medium

Find the particular solution to [formula] with [formula].

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