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[
{
"qid": "part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_9402_single_15",
"messages": [
{
"content": "<image> Analyze the image. Extract and output only the LaTeX formulas present in the image, in LaTeX code format. Ignore inline formulas, all other text, and do not include any explanations.",
"role": "user"
},
{
"content": "\\begin{align}\n\\frac{d}{dt}X_1+\\frac{d}{dt}X_2 &=\\left(AX_1+B\\begin{bmatrix}c_1\\cos{\\omega_1t}\\\\ d_1\\cos{\\omega_1t}\\end{bmatrix}\\right)+ \\left(AX_2+B\\begin{bmatrix}c_2\\cos{\\omega_2t}\\\\ d_2\\cos{\\omega_2t}\\end{bmatrix}\\right) \\\\ &=AX_1+AX_2+B\\begin{bmatrix}c_1\\cos{\\omega_1t}\\\\ d_1\\cos{\\omega_1t}\\end{bmatrix}+B\\begin{bmatrix}c_2\\cos{\\omega_2t}\\\\ d_2\\cos{\\omega_2t}\\end{bmatrix} \\\\ &= A(X_1+X_2) + B\\left(\\begin{bmatrix}c_1\\cos{\\omega_1t}\\\\ d_1\\cos{\\omega_1t}\\end{bmatrix}+\\begin{bmatrix}c_2\\cos{\\omega_2t}\\\\ d_2\\cos{\\omega_2t}\\end{bmatrix}\\right) \\\\ &= AX+B\\begin{bmatrix}u_1(t) \\\\ u_2(t)\\end{bmatrix} \\\\ &=\\frac{d}{dt}X.\n\\end{align}",
"role": "assistant"
}
],
"images": [
"./asset/images/question_part-00014-8bdd594e-bf2f-4dee-8277-9c33e8468d28-c000_9402_single_15.png"
],
"model_answer": "\\begin{align}\n\\frac{d}{dt}X_1 + \\frac{d}{dt}X_2 &= \\left(AX_1 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix}\\right) + \\left(AX_2 + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX_1 + AX_2 + B\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + B\\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\\\ &= A(X_1 + X_2) + B\\left(\\begin{bmatrix}c_1\\cos\\omega_1 t\\\\d_1\\cos\\omega_1 t\\end{bmatrix} + \\begin{bmatrix}c_2\\cos\\omega_2 t\\\\d_2\\cos\\omega_2 t\\end{bmatrix}\\right)\\\\ &= AX + B\\begin{bmatrix}u_1(t)\\\\u_2(t)\\end{bmatrix}\\\\ &= \\frac{d}{dt}X.\n\\end{align}",
"edit_acc": 0.1109350237717908
}
] |