Ordinary Differential Equations

Recall a first-order ODE is called linear if it can be written in the form (1.1.4), where it is called homogeneous whenever \(g(t)=0\) for all \(t\) and nonohomogeneous whenenever there exists some \(t\) such that \(g(t)\neq 0\text{.}\) Notice that if (1.1.4) is homogeneous, then it is separable and if (1.1.4) is not homogeneous, then it is not separable. However, there always exists a function \(\mu\text{,}\) called the integrating factor that when multiplied to both sides allows us to integrate both sides with respect to \(t\text{.}\)

Theorem 1.11.1.

The integrating factor for (1.1.4) is \(\mu(t)=e^{\int P(t)\mathrm{d}t}\text{.}\)

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Proof.

later!

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Algorithm 1.11.2. Integrating factor method.

To solve (1.1.4), we do the following steps.

1. Put (1.1.4) into standard form.
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2. Find the integrating factor \(\mu\text{.}\)
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3. Multiply both sides of (1.1.4) by \(\mu\text{.}\)
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4. Rewrite the left-hand side using the product rule as \(\dfrac{\mathrm{d}}{\mathrm{d}t} \Big[ \mu(t)y \Big]\text{,}\) which will always happen if you computed \(\mu\) correctly.
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5. Integrate both sides with respect to \(t\text{.}\)
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6. Solve the resulting equation for \(y\text{.}\)
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7. If you have an initial condition, then use it to find the constant of integration \(C\)
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It is important to note that Algorithm 1.11.2 can only be done on equations of form (1.1.4). All solutions \(y\) of (1.1.4) take the form \(y=y_h+y_p\text{,}\) where \(y_h\) is the homogeneous solution, i.e. the solution of the homogeneous analogue of equation (1.1.4) obtained when the right-hand side is set equal to zero for all \(t\) and \(y_p\) solves (1.1.4) itself. This occurs because of the nature of the homogeneous solution: it disappears when substituted into the ODE (think about why that is)!

Example 1.11.3.

Consider the initial value problem \((t+1)y'-ty=\dfrac{e^t}{1+t^2}, \quad y(0)=3\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)automous, and (non)separable?
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2. What method(s) can you use to solve this differential equation?
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3. Solve the initial value problem.
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Example 1.11.4.

Consider the differential equation \(\cos(t)y'-2\sin(t)y=\sec^3(t)\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)automous, and (non)separable?
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2. What method(s) can you use to solve this differential equation?
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3. Find an explicit solution to the differential equation.
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Example 1.11.5.

Consider the differential equation \(ty'+(t-4)y=8t^5e^{3t}\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)automous, and (non)separable?
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2. What method(s) can you use to solve this differential equation?
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3. Find an explicit solution to the differential equation.
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Example 1.11.6.

Recall Section 1.6. A \(1200 \mathrm{gal}\) tank is filled with \(600 \mathrm{gal}\) of pure water. Brine containing \(5 \dfrac{\mathrm{lb}}{\mathrm{gal}}\) of salt is pumped into the tank at a rate of \(6 \dfrac{\mathrm{gal}}{\mathrm{min}}\text{.}\) The well mixed solution is then pumped out at a rate of \(3 \dfrac{\mathrm{gal}}{\mathrm{min}}\text{.}\) Let \(A(t)\) represent the amount of salt in the tank.

1. Set up the IVP. Find the order of the differential equation. Is the differential equation (non)linear, (non)autonomous, (non)separable?
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2. What method(s) can you use to solve this differential equation?
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3. Solve the initial value problem.
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