How to solve this math problem about convolution?

1. 📝 Identify: f(t) = e^(1t), g(t) = e^(5t) 2. 🧮 Convolution Formula: (f * g)(t) = ∫₀ᵗ f(τ)·g(t-τ) dτ = ∫₀ᵗ e^(1τ)·e^(5(t-τ)) dτ

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📐 MathLevel 2 — 🟢 Basicconvolution👁 24 views⏱ 1 min read

Solve: convolution e^(1t) * e^(5t)

📝 Problem

convolution e^(1t) * e^(5t)

1. 📝 Identify: ,

2. 🧮 Convolution Formula: (f * g)(t) = ∫₀ᵗ f(τ)·g(t-τ) dτ = ∫₀ᵗ e^(1τ)·e^(5(t-τ)) dτ

3. 📐 Evaluate integral: ∫₀ᵗ e^(1τ + 5τ) dτ = e^(5t) ∫₀ᵗ e^(-4τ) dτ

= e^(5t) · [e^(-4τ)/-4]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^(1τ)·e^(5(t-τ)) dτ

3. 📐 Evaluate integral: ∫₀ᵗ e^(1τ + 5τ) dτ = e^(5t) ∫₀ᵗ e^(-4τ) dτ

= e^(5t) · [e^(-4τ)/-4]₀ᵗ

🔒 6 steps remaining

📐 Lv.2

Solve: Compute the convolution: e^(2t) * e^(5t)