Differential Equations Sample: Second-Order ODE Solved in Handwriting

A full worked example of a second-order linear ODE with initial conditions, rendered step by step by Scrawl AI.

The problem

Consider a standard second-order linear ODE with constant coefficients and initial conditions:

y'' + 3y' + 2y = e^x, y(0) = 1, y'(0) = 0

What the handwritten solution shows

Scrawl walks through every step a student would write by hand:

Characteristic equation. r² + 3r + 2 = 0 factors into (r+1)(r+2), giving homogeneous solution y_h = C₁e^(-x) + C₂e^(-2x).
Particular solution. Try y_p = Ae^x, substitute, solve A = 1/6, giving y_p = (1/6)e^x.
General solution. y = C₁e^(-x) + C₂e^(-2x) + (1/6)e^x.
Apply initial conditions.y(0) = 1 and y'(0) = 0 give a 2×2 system for C₁ and C₂. Solution: y = (2/3)e^(-x) + (1/6)e^(-2x) + (1/6)e^x.

Laplace transform alternative

Scrawl can also solve the same IVP using Laplace transforms: L{y} = Y(s), apply to both sides, solve algebraically for Y(s), then take the inverse transform using a partial fraction decomposition. The final answer matches the characteristic equation method. Both paths are rendered in handwriting with proper fraction bars and derivative notation.

Why students submit it

Differential equations homework demands clean presentation — derivative primes, integral signs, e^(-x) notation — that chatbots render poorly in plain text. Scrawl outputs a downloadable handwritten PDF with every glyph naturally rendered.

Try it on your own ODE

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