Differential Equations

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A differential equation is an equation involving derivatives of the dependent variable with respect to independent variable(variables). Example: dy/dx + y = 0 or x(dy/dx) + y = 0. It contains variable

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Ordinary differential equation: involves derivatives with respect to only ONE independent variable. Example: d²y/dx² + dy/dx = 0. Partial differential equation: involves derivatives with respect to MO

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Order is the order of the highest order derivative in the differential equation. Examples: (1) dy/dx = e^x has order 1, (2) d²y/dx² + y = 0 has order 2, (3) d³y/dx³ + x²(d²y/dx²)³ = 0 has order 3. Ord

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Degree is the highest power (positive integral index) of the highest order derivative when the differential equation is a polynomial in derivatives. Example: (dy/dx)² + dy/dx = sin²y has degree 2. Deg

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Order = 3 (highest derivative is d³y/dx³), Degree = 1 (the power of highest order derivative d³y/dx³ is 1). Note that (d²y/dx²)² has power 2, but d²y/dx² is not the highest order derivative.

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General solution: Contains arbitrary constants equal to the order of the differential equation. Example: y = a sin(x + b) where a, b are arbitrary constants. Particular solution: Free from arbitrary c

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The curve y = φ(x) is called the solution curve (integral curve) of the given differential equation when the function φ(x) satisfies the differential equation. When φ(x) and its derivatives are substi

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A first order differential equation dy/dx = F(x,y) where F(x,y) can be expressed as F(x,y) = g(x)h(y), i.e., dy/dx = g(x)h(y). Here g(x) depends only on x and h(y) depends only on y. Variables can be