MATH1012 - Lecture 28

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MATH1012 - Lecture 27
Skeleton Slides

Second-Order Linear Differential Equations

• These are differential equations which can be written in the form: $\frac{d^{2}y}{d{x}^2}+p(x)\frac{dy}{dx}+q(x)y=g(x)$
• If $g(x)=0$ it is homogeneous, and if $g(x)\ne 0$ it is inhomogeneous

Principle of Superposition

If $y_1(x)$ and $y_2(x)$ are solutions of a second order linear homogeneous differential equation, then so is $c_1{y}_1(x)+c_2{y}_2(x)$ for any $c_1$, $c_2$

i.e. $y(x)=c_1{y}_1(x)+c_2{y}_2(x)$ gives all solutions

• If coefficients of the derivatives are constant we are given solutions: $y(x)=e^{mx}$
• We are given: $m=\frac{-p\pm \sqrt{p^2-4q}}{2}$
  • $m$ is found in a quadratic equation which can be solved using the quadratic formula
  • See slides for more info
• The determinant of this equations is $p^2-4q$
  • We must consider what happens for different cases

Cases for Determinant

1. Determinant $>0$
   • The characteristic equation has two distinct real roots $m_1,m_2,$ yielding solutions $y_1(x)=e^{m_{1}x}$ and $y_2(x)=e^{m_{2}x}$
   • Hence general solution: $y(x)=c_1{e}^{m_{1}x}+c_2{e}^{m_{2}x}$
2. Determinant $<0$
   • The characteristic equation will have complex conjugate roots, which we can write as $m=a\pm ib$ where $i^2=-1$
   • These yield solutions $y_1(x)=e^{ax}\cos (bx)$ and $y_2(x)=e^{ax}\sin (bx)$
   • Hence general solution: $y(x)=c_1{e}^{ax}\cos (bx)+c_2{e}^{ax}\sin (bx)$
3. Determinant $=0$
   • The characteristic equation has a single (repeated) root $m=-\frac{p}{2}$
   • The general solution: $y(x)=c_1{e}^{mx}+c_{2}x{e}^{mx}$

Initial and Boundary Value

• A second order differential equation is usually accompanied by two extra conditions on the unknown function
  • Initial value problem: conditions are given at the same point
    • e.g. subject to $y(0)=1$ and $\frac{dy}{dx}(0)=3$
  • Boundary value problem: conditions are given at different points
    • e.g. subject to $y(0)=0$ and $y(1)=2$
• Initial and boundary conditions are referred to as auxiliary conditions

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MATH1012 - Lecture 29
Annotated Slides