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Question

The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\)

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

Answer

1. Try ansatz: \(y=e^{mx}\).

2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\)

3. Using the solution of the auxiliary equation to solve the ODE.

Question

The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\)

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

Answer

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Question

The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\)

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: [Method Steps]

Answer

1. Try ansatz: \(y=e^{mx}\).

2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\)

3. Using the solution of the auxiliary equation to solve the ODE.

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**The solution method for 2nd order linear homogeneous ODE**

> The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\) If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: 1. Try ansatz: \(y=e^{mx}\). 2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\) 3. Using the solution of the auxiliary equation to solve the ODE. <span>

> The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\) If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions: 1. Try ansatz: \(y=e^{mx}\). 2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\) 3. Using the solution of the auxiliary equation to solve the ODE. <span>

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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