A differential equation is an equation that involves an unknown function and one or more of its derivatives. Differential equations model real-world phenomena like population growth, radioactive decay, heat flow, and more. In Class 12, we study ordinary differential equations (ODEs) involving a single independent variable.
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Key Definitions
- Order: The order of a differential equation is the order of the highest derivative present.
- Degree: The degree is the power of the highest-order derivative, after the equation is made free of radicals and fractions in the derivatives.
- General Solution: A solution containing arbitrary constants (equal in number to the order).
- Particular Solution: Obtained by substituting specific initial/boundary conditions.
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Methods of Solving
1. Variable Separable Method:
If a differential equation can be written as f(y) dy = g(x) dx, integrate both sides separately.
2. Homogeneous Differential Equations:
If dy/dx = F(y/x), substitute y = vx (so dy/dx = v + x dv/dx), then separate variables.
3. Linear Differential Equations:
Form: dy/dx + P(x) y = Q(x)
Integrating Factor (IF) = eintegral of P dx
Solution: y × IF = integral of [Q × IF] dx + C
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Worked Examples
Find the order and degree of: (d2y/dx2)3 + (dy/dx)2 + y = 0.
Highest derivative is d2y/dx2 (second derivative). Order = 2. Power of the highest derivative term is 3. Degree = 3.
Solve: dy/dx = (1 + y2)/(1 + x2)
Separating: dy/(1 + y2) = dx/(1 + x2)
Integrating both sides: arctan(y) = arctan(x) + C
=> arctan(y) - arctan(x) = C
Solve the homogeneous equation dy/dx = (x + y)/x.
Write dy/dx = 1 + y/x. Let y = vx => dy/dx = v + x dv/dx.
v + x dv/dx = 1 + v => x dv/dx = 1 => dv = dx/x
Integrating: v = ln|x| + C => y/x = ln|x| + C => y = x ln|x| + Cx
Solve dy/dx + y/x = x2.
This is linear. P = 1/x, Q = x2.
IF = eintegral of 1/x dx = eln x = x.
Solution: y × x = integral of x × x2 dx = integral of x3 dx = x4/4 + C
=> xy = x4/4 + C
Find the particular solution of dy/dx = y tan x, given y = 1 when x = 0.
Separating: dy/y = tan x dx => ln|y| = ln|sec x| + C => y = A sec x.
At x = 0, y = 1: 1 = A sec 0 = A. So y = sec x.
Verify that y = e2x is a solution of y'' - 3y' + 2y = 0.
y = e2x, y' = 2e2x, y'' = 4e2x.
y'' - 3y' + 2y = 4e2x - 6e2x + 2e2x = 0. Verified.
Form the differential equation of y = A sin x + B cos x (A, B arbitrary).
y' = A cos x - B sin x; y'' = -A sin x - B cos x = -y.
So the differential equation is y'' + y = 0.
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Common mistakes
- Confusing order and degree: Order is the highest derivative's order; degree requires the equation to be polynomial in derivatives first.
- Missing constant of integration: Every integration step in solving a DE must include a constant C; missing it leads to a wrong general solution.
- Incorrect integrating factor: For linear DEs, the IF is eintegral of P dx — do not include C in this step.
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Summary
Differential equations are classified by order and degree. The three main solution methods are variable separable, homogeneous substitution (y = vx), and the integrating factor method for linear DEs. Always verify solutions by substituting back into the original equation.