y'''+4y''+7y=0

asked by guest
on Nov 27, 2024 at 3:52 am



You asked:

Solve the equation \(y^{\prime \prime \prime} + 4 \cdot y^{\prime \prime} + 7 y = 0\) for the variable \(y\).

MathBot Answer:

\[y{\left(t \right)} = C_{3} e^{- \frac{\left(8 + 2^{\frac{2}{3}} \sqrt[3]{317 + 3 \sqrt{9345}} + \frac{32 \cdot \sqrt[3]{2}}{\sqrt[3]{317 + 3 \sqrt{9345}}}\right) t}{6}} + C_{1} e^{\frac{\left(-16 + 2^{\frac{2}{3}} \sqrt[3]{317 + 3 \sqrt{9345}} + \frac{32 \cdot \sqrt[3]{2}}{\sqrt[3]{317 + 3 \sqrt{9345}}}\right) t}{12}} \sin{\left(\frac{\sqrt[3]{2} \sqrt{3} \left(\frac{32}{\sqrt[3]{317 + 3 \sqrt{9345}}} - \sqrt[3]{2} \sqrt[3]{317 + 3 \sqrt{9345}}\right) t}{12} \right)} + C_{2} \cos{\left(\frac{\sqrt[3]{2} \sqrt{3} \left(\frac{32}{\sqrt[3]{317 + 3 \sqrt{9345}}} - \sqrt[3]{2} \sqrt[3]{317 + 3 \sqrt{9345}}\right) t}{12} \right)} e^{\frac{\left(-16 + 2^{\frac{2}{3}} \sqrt[3]{317 + 3 \sqrt{9345}} + \frac{32 \cdot \sqrt[3]{2}}{\sqrt[3]{317 + 3 \sqrt{9345}}}\right) t}{12}}\]

\(C_{1}\), \(C_{2}\), \(C_{3}\) are arbitrary constants.

\(e\) is Euler's number, a mathematical constant that is approximately \(2.71828\).

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