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Systems of Differential Equations Calculator

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Example
Created on 2024-06-20Asked by William Taylor (Solvelet student)
Solve the system of differential equations: {x˙=3x+4yy˙=−4x+3y \begin{cases} \dot{x} = 3x + 4y \\ \dot{y} = -4x + 3y \end{cases}

Solution

To solve the system of differential equations: {x˙=3x+4yy˙=−4x+3y \begin{cases} \dot{x} = 3x + 4y \\ \dot{y} = -4x + 3y \end{cases} 1. **Find the eigenvalues of the coefficient matrix:** A=(34−43). A = \begin{pmatrix} 3 & 4 \\ -4 & 3 \end{pmatrix}. 2. **Characteristic equation:** det⁡(A−λI)=(3−λ)2+16=0  ⟹  λ2−6λ+25=0. \det(A - \lambda I) = (3 - \lambda)^2 + 16 = 0 \implies \lambda^2 - 6\lambda + 25 = 0. λ=3±4i. \lambda = 3 \pm 4i. 3. **Eigenvectors:** For λ=3+4i\lambda = 3 + 4i: (3−(3+4i)4−43−(3+4i))(v1v2)=0. \begin{pmatrix} 3 - (3 + 4i) & 4 \\ -4 & 3 - (3 + 4i) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = 0. 4. **General solution:** (x(t)y(t))=e3t[c1(cos⁡(4t)−sin⁡(4t))+c2(sin⁡(4t)cos⁡(4t))]. \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = e^{3t} \left[ c_1 \begin{pmatrix} \cos(4t) \\ -\sin(4t) \end{pmatrix} + c_2 \begin{pmatrix} \sin(4t) \\ \cos(4t) \end{pmatrix} \right]. 5. **Result:** The solution to the system is: {x(t)=e3t(c1cos⁡(4t)+c2sin⁡(4t))y(t)=e3t(−c1sin⁡(4t)+c2cos⁡(4t)). \begin{cases} x(t) = e^{3t} (c_1 \cos(4t) + c_2 \sin(4t)) \\ y(t) = e^{3t} (-c_1 \sin(4t) + c_2 \cos(4t)) \end{cases}. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Amelia Anderson on Solvelet
1. Solve the system of differential equations dxdt=3x+2y\frac{dx}{dt} = 3x + 2y, dydt=−2x+y\frac{dy}{dt} = -2x + y with initial conditions x(0)=1x(0) = 1, y(0)=−1y(0) = -1.2. Determine if the system of differential equations dx/dt=x+2y dx/dt = x + 2y , dy/dt=−2x+3y dy/dt = -2x + 3y has a unique solution.,
DefinitionDescription of systems of differential equations: simply, a system of differential equations if a case when we have some partial differential equations which together describe behaviour of several related variables. These systems are solved using matrix algebra, eigenvalues, numerical techniques(^)( For instance, the system dtdx​=3x+4y,dtdy​=−x+y
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