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Calculus - Differential equations (HL)

Grade 11IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A differential equation is an equation involving a function and its derivatives. In IB AI HL, the focus is on first-order differential equations of the form dydx=f(x,y)\frac{dy}{dx} = f(x, y).

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Separation of Variables: A technique used when the equation can be written as dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y). We rearrange it to ∫1h(y)dy=∫g(x)dx\int \frac{1}{h(y)} dy = \int g(x) dx to find the general solution.

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Euler's Method: A numerical method used to approximate the solution to a differential equation at specific points. It uses a step size hh to iterate from an initial condition (x0,y0)(x_0, y_0).

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Slope Fields: A graphical representation where small line segments are drawn at points (x,y)(x, y) on a coordinate plane, with the slope equal to the value of dydx\frac{dy}{dx} at that point.

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Coupled Linear Differential Equations: Systems of the form dxdt=ax+by\frac{dx}{dt} = ax + by and dydt=cx+dy\frac{dy}{dt} = cx + dy. These can be represented using matrices as x˙=Ax\mathbf{\dot{x}} = \mathbf{Ax}.

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Phase Portraits: Graphs used to visualize the behavior of coupled systems by plotting yy against xx. They help identify equilibrium points and determine if they are stable (sinks), unstable (sources), or saddle points.

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Second-order Differential Equations: In AI HL, these are often converted into a system of two first-order equations to be solved using matrix methods.

📐Formulae

Separation of Variables: ∫1g(y)dy=∫f(x)dx\text{Separation of Variables: } \int \frac{1}{g(y)} dy = \int f(x) dx

Euler’s Method: xn+1=xn+h\text{Euler's Method: } x_{n+1} = x_n + h

Euler’s Method: yn+1=yn+h×f(xn,yn)\text{Euler's Method: } y_{n+1} = y_n + h \times f(x_n, y_n)

Coupled System: (dxdtdydt)\text{Coupled System: } \begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{pmatrix} = (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$$

Characteristic Equation for Eigenvalues: det⁡(A−λI)=0\text{Characteristic Equation for Eigenvalues: } \det(\mathbf{A} - \lambda \mathbf{I}) = 0

💡Examples

Problem 1:

Solve the differential equation dydx=2xy\frac{dy}{dx} = \frac{2x}{y} given the initial condition y(0)=4y(0) = 4.

Solution:

∫y dy=∫2x dx\int y \, dy = \int 2x \, dx y22=x2+C\frac{y^2}{2} = x^2 + C Using (0,4)(0, 4): 422=02+C  ⟹  8=C\frac{4^2}{2} = 0^2 + C \implies 8 = C y22=x2+8  ⟹  y2=2x2+16\frac{y^2}{2} = x^2 + 8 \implies y^2 = 2x^2 + 16 y=2x2+16y = \sqrt{2x^2 + 16}

Explanation:

We use the separation of variables method. We move all yy terms to the left and xx terms to the right, integrate both sides, and solve for the constant CC using the initial condition.

Problem 2:

Use Euler's method with a step size of h=0.1h = 0.1 to approximate y(0.2)y(0.2) for the differential equation dydx=x+y\frac{dy}{dx} = x + y, starting at (0,1)(0, 1).

Solution:

Step 1: x0=0,y0=1,f(x,y)=x+yx_0 = 0, y_0 = 1, f(x, y) = x + y y1=y0+h(x0+y0)=1+0.1(0+1)=1.1y_1 = y_0 + h(x_0 + y_0) = 1 + 0.1(0 + 1) = 1.1 x1=0+0.1=0.1x_1 = 0 + 0.1 = 0.1 Step 2: y2=y1+h(x1+y1)=1.1+0.1(0.1+1.1)=1.1+0.1(1.2)=1.22y_2 = y_1 + h(x_1 + y_1) = 1.1 + 0.1(0.1 + 1.1) = 1.1 + 0.1(1.2) = 1.22 x2=0.1+0.1=0.2x_2 = 0.1 + 0.1 = 0.2 So, y(0.2)≈1.22y(0.2) \approx 1.22.

Explanation:

Euler's method approximates the next yy-value by moving along the tangent line. We perform two iterations since we need to reach x=0.2x = 0.2 from x=0x = 0 with a step of 0.10.1.

Problem 3:

Find the eigenvalues of the coupled system: dxdt=4x−2y\frac{dx}{dt} = 4x - 2y dydt=x+y\frac{dy}{dt} = x + y

Solution:

The matrix A\mathbf{A} is: A=(4−211)\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix} Find det⁡(A−λI)=0\det(\mathbf{A} - \lambda \mathbf{I}) = 0: ∣4−λ−211−λ∣=0\begin{vmatrix} 4-\lambda & -2 \\ 1 & 1-\lambda \end{vmatrix} = 0 (4−λ)(1−λ)−(−2)(1)=0(4-\lambda)(1-\lambda) - (-2)(1) = 0 4−4λ−λ+λ2+2=04 - 4\lambda - \lambda + \lambda^2 + 2 = 0 λ2−5λ+6=0\lambda^2 - 5\lambda + 6 = 0 (λ−2)(λ−3)=0(\lambda - 2)(\lambda - 3) = 0 λ1=2,λ2=3\lambda_1 = 2, \lambda_2 = 3

Explanation:

To analyze the stability or find the general solution of a coupled system, we first find the eigenvalues of the coefficient matrix. Here, both eigenvalues are positive, indicating the equilibrium at (0,0)(0,0) is an unstable source.