How to solve this math problem about convolution?

1. 📝 Identify: f(t) = e^(2t), g(t) = e^(4t) 2. 🧮 Convolution Formula: (f * g)(t) = ∫₀ᵗ f(τ)·g(t-τ) dτ = ∫₀ᵗ e^(2τ)·e^(4(t-τ)) dτ

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📐 MathLevel 2 — 🟢 Basicconvolution👁 17 views⏱ 1 min read

Solve: Compute the convolution: e^(2t) * e^(4t)

📝 Problem

Compute the convolution: e^(2t) * e^(4t)

1. 📝 Identify: ,

2. 🧮 Convolution Formula: (f * g)(t) = ∫₀ᵗ f(τ)·g(t-τ) dτ = ∫₀ᵗ e^(2τ)·e^(4(t-τ)) dτ

3. 📐 Evaluate integral: ∫₀ᵗ e^(2τ + 4τ) dτ = e^(4t) ∫₀ᵗ e^(-2τ) dτ

= e^(4t) · [e^(-2τ)/-2]₀ᵗ

🔒 6 steps remaining

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1. 📝 Identify: ,

2. 🧮 Convolution Formula: (f * g)(t) = ∫₀ᵗ f(τ)·g(t-τ) dτ = ∫₀ᵗ e^(2τ)·e^(4(t-τ)) dτ

3. 📐 Evaluate integral: ∫₀ᵗ e^(2τ + 4τ) dτ = e^(4t) ∫₀ᵗ e^(-2τ) dτ

= e^(4t) · [e^(-2τ)/-2]₀ᵗ

🔒 6 steps remaining

📐 Lv.2

Solve: convolution e^(3t) * e^(4t)