01 · Explore
Introduction and Basic Concepts
A differential equation is an equation that involves the derivative of a dependent variable with respect to one or more independent variables.
In previous studies of calculus, we learned how to find the derivative of a function f(x). In differential equations, we work in reverse: given an equation involving derivatives, we seek to find the original function y = f(x) that satisfies the relationship. For example, dy/dx = g(x) is a simple differential equation where we seek a function y whose slope at any point x is g(x).
An ordinary differential equation (ODE) involves derivatives with respect to only one independent variable. For instance, 2(d²y/dx²) + (dy/dx)³ = 0 is an ODE because y is differentiated only with respect to x. If an equation involves derivatives with respect to multiple independent variables, it is called a partial differential equation, though Class 12 mathematics focuses exclusively on ODEs.
Common notations used for derivatives include y' for dy/dx, y'' for d²y/dx², and y''' for d³y/dx³. For higher orders, we use yₙ to denote the nth derivative dⁿy/dxⁿ. These equations are vital in fields like Physics, Chemistry, and Economics to model rates of change.
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Question
Identify the independent and dependent variables in the equation t(d²s/dt²) + s = 0.
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02 · Explore
Order and Degree of a Differential Equation
Every differential equation is characterized by its order and, if defined, its degree.
The order of a differential equation is defined as the order of the highest-order derivative present in the equation. For example, in the equation (d³y/dx³) + x²(d²y/dx²)³ = 0, the highest derivative is of the third order, so the order of the differential equation is 3.
The degree of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial equation in its derivatives (y', y'', etc.). If the derivatives are part of a transcendental function, such as sin(y') or eʸ'', the degree is not defined. It is important to note that both order and degree must be positive integers.
To find the degree, first ensure the equation is free from radicals and fractions regarding the derivatives. Then, identify the highest-order derivative and look at its exponent. In (d²y/dx²)³ + (dy/dx)² + sin(y) = 0, the highest order derivative is d²y/dx², so the order is 2. The power of this highest order derivative is 3, so the degree is 3. However, in (d²y/dx²) + sin(dy/dx) = 0, the order is 2 but the degree is not defined because of the sin(dy/dx) term.
| Differential Equation | Highest Derivative | Order | Degree |
|---|---|---|---|
| dy/dx - cos x = 0 | dy/dx | 1 | 1 |
| xy(d²y/dx²) + x(dy/dx)² - y(dy/dx) = 0 | d²y/dx² | 2 | 1 |
| y''' + y² + eʸ' = 0 | y''' | 3 | Not Defined |
| (d⁴y/dx⁴) + sin(y''') = 0 | d⁴y/dx⁴ | 4 | Not Defined |
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Question
Find the order and degree of (y'')² + (y')³ + sin(y) = 0.
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03 · Explore
General and Particular Solutions
A solution of a differential equation is a function that satisfies the equation when substituted into it.
A general solution (or primitive) is a solution that contains as many arbitrary constants as the order of the differential equation. For a second-order equation, the general solution will have two constants, typically denoted as A and B or C₁ and C₂. These constants represent a family of curves.
A particular solution is obtained from the general solution by assigning specific values to the arbitrary constants based on given conditions (initial or boundary conditions). A particular solution contains no arbitrary constants and represents a single specific curve from the family.
To verify if a function is a solution, differentiate the function as many times as the order of the differential equation and substitute the function and its derivatives into the equation. If the Left Hand Side (LHS) equals the Right Hand Side (RHS), the function is a valid solution.
Verification of a Solution
Verify y = e⁻³ˣ is a solution of (d²y/dx²) + (dy/dx) - 6y = 0
1. Find derivatives: y' = -3e⁻³ˣ and y'' = 9e⁻³ˣ. 2. Substitute into LHS: 9e⁻³ˣ + (-3e⁻³ˣ) - 6(e⁻³ˣ) = 9e⁻³ˣ - 9e⁻³ˣ = 0. 3. Since LHS = RHS, it is a solution.
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Question
How many arbitrary constants are in the particular solution of a 4th order differential equation?
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04 · Explore
Method of Variable Separation
This method is used when the differential equation can be rearranged so that all terms involving y are on one side with dy, and all terms involving x are on the other with dx.
A first-order differential equation dy/dx = F(x, y) is of the variable separable type if F(x, y) can be expressed as g(x)h(y). By rearranging, we get [1/h(y)] dy = g(x) dx. We then integrate both sides to find the general solution: ∫ [1/h(y)] dy = ∫ g(x) dx + C.
This technique is often used in growth and decay problems. For instance, if the rate of change of a quantity P is proportional to P itself (dP/dt = kP), we can separate variables as (1/P) dP = k dt. Integrating gives log|P| = kt + C, which leads to the exponential growth model P = Aeᵏᵗ.
When solving, always remember to add the constant of integration 'C' immediately after integrating. If a point (x₀, y₀) is provided, substitute these values into the general solution to find the specific value of C for the particular solution.
Solving by Variable Separation
Solve dy/dx = (1 + y²) / (1 + x²)
1. Separate variables: dy / (1 + y²) = dx / (1 + x²). 2. Integrate both sides: ∫ dy / (1 + y²) = ∫ dx / (1 + x²). 3. Apply standard integrals: tan⁻¹ y = tan⁻¹ x + C. This is the general solution.
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Question
Solve dy/dx = -4xy² given y = 1 when x = 0.
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05 · Explore
Homogeneous Differential Equations
A differential equation dy/dx = F(x, y) is homogeneous if F(x, y) is a homogeneous function of degree zero.
A function F(x, y) is homogeneous of degree n if F(λx, λy) = λⁿ F(x, y). For a differential equation to be homogeneous, the degree n must be zero, meaning F(λx, λy) = F(x, y). This usually happens when every term in the numerator and denominator has the same combined power of x and y.
To solve these equations, we use the substitution y = vx. Differentiating this with respect to x using the product rule gives dy/dx = v + x(dv/dx). Substituting these into the original equation transforms it into a variable separable equation in terms of v and x.
After separating variables and integrating, the final step is to replace v with y/x to express the solution in terms of the original variables x and y. If the equation is of the form dx/dy = F(x, y), use the substitution x = vy instead.
Steps to Solve Homogeneous Equations
- 1
Verify Homogeneity
Check if F(λx, λy) = F(x, y) or if all terms have the same degree.
- 2
Substitute
Let y = vx, then dy/dx = v + x(dv/dx).
- 3
Separate and Integrate
Rearrange to separate v and x terms, then integrate both sides.
- 4
Back-Substitute
Replace v with y/x to get the final general solution.
The sequential process for solving first-order homogeneous differential equations.
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Question
Is dy/dx = (x² + y²) / xy homogeneous? If so, what is the degree of the function?
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06 · Explore
Linear Differential Equations
A first-order linear differential equation is of the form dy/dx + Py = Q, where P and Q are constants or functions of x only.
To solve this type of equation, we use an 'Integrating Factor' (I.F.). The I.F. is defined as e raised to the power of the integral of P with respect to x: I.F. = e^∫P dx. Multiplying the entire differential equation by this factor makes the left side a perfect derivative of (y × I.F.).
The general solution is then given by the formula: y × (I.F.) = ∫ (Q × I.F.) dx + C. This method effectively converts a linear relationship of derivatives into a standard integration problem.
A second form of linear equation is dx/dy + P₁x = Q₁, where P₁ and Q₁ are functions of y only. In this case, the Integrating Factor is I.F. = e^∫P₁ dy, and the solution is x × (I.F.) = ∫ (Q₁ × I.F.) dy + C. Always ensure the coefficient of dy/dx (or dx/dy) is 1 before identifying P and Q.
Solving a Linear Equation
Solve dy/dx + y/x = x²
1. Identify P = 1/x and Q = x². 2. Find I.F. = e^∫(1/x)dx = e^(log x) = x. 3. Apply formula: y(x) = ∫(x² * x) dx + C. 4. Integrate: xy = ∫x³ dx + C => xy = x⁴/4 + C. 5. General solution: y = x³/4 + C/x.
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Question
What is the Integrating Factor for dy/dx - y = cos x?
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07 · Explore
Applications of Differential Equations
Differential equations are used to model real-world phenomena involving rates of change, such as population growth, interest rates, and cooling.
In banking, if interest is compounded continuously, the rate of change of principal P with respect to time t is dP/dt = rP/100, where r is the annual interest rate. This is a variable separable equation that results in the exponential growth of money.
In biology, the growth of a bacteria culture often follows the rule that the rate of increase is proportional to the number of bacteria present (dN/dt = kN). By solving this, we can predict the population at any future time t given the initial count.
Geometric applications include finding the equation of a curve when the slope of its tangent (dy/dx) is given as a function of x and y. For example, if the slope is always equal to the sum of the coordinates, we solve dy/dx = x + y, which is a linear differential equation.
Continuous Compounding
dP/dt = 0.05P; P(0) = 1000
1. Separate: dP/P = 0.05 dt. 2. Integrate: log P = 0.05t + C. 3. Exponential form: P = Ae^(0.05t). 4. Use P(0)=1000: 1000 = A * e⁰ => A = 1000. 5. Result: P = 1000e^(0.05t). To double (P=2000): 2 = e^(0.05t) => t = (log 2)/0.05 = 20 log 2 years.
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Question
If the slope of a tangent to a curve at (x, y) is 2x/y², write the differential equation.
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