solving-second-order-differential-equations

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Second order differential equations

A second order differential equation is an equation containing second order derivatives (and possibly first order derivatives) but no higher order derivatives
- For example $fraction numerator d squared y over denominator d x squared end fraction equals 12 x$ is a second order differential equation
- And so is $fraction numerator d squared x over denominator d t squared end fraction minus 5 fraction numerator d x over denominator d t end fraction plus 7 x equals 5 sin t$
- But $fraction numerator d cubed y over denominator d x cubed end fraction plus 3 fraction numerator d squared y over denominator d x squared end fraction minus 2 fraction numerator d y over denominator d x end fraction equals 0$ is not, because it contains the third order derivative $fraction numerator d cubed y over denominator d x cubed end fraction$

We divide second order differential equations into two main types
A homogeneous second order differential equation is of the form <img alt=”a fraction numerator d squared y over denominator d x squared end fraction plus b fraction numerator d y over denominator d x end fraction plus c y equals 0″ data-mathml='<math ><semantics><mrow><mi>a</mi><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>b</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>c</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding=”a