Unit 3: Linear differential equation-II

I. Orientation — governing framework

A linear differential equation combines an unknown function and its derivatives only to the first power, with coefficients independent of the unknown function. In this unit, the main problem is to solve a non-homogeneous equation with constant coefficients or an Euler–Cauchy equation, usually by finding a complementary function and a particular integral.

• General linear form: A second-order equation may be written as
  

  TEXT

    a₂(x)y'' + a₁(x)y' + a₀(x)y = f(x)

  
  where y is the dependent variable, x is the independent variable, and f(x) is the forcing function.

• Homogeneous and non-homogeneous parts: If f(x)=0, the equation is homogeneous; if f(x)≠0, it is non-homogeneous.

• Linearity assumption: Terms such as y², yy', or sin y are not allowed in a linear equation.

• Superposition principle: For a linear operator L, if L[y₁]=f₁ and L[y₂]=f₂, then L[c₁y₁+c₂y₂]=c₁f₁+c₂f₂.

• Complete solution: The solution is expressed as
  

  TEXT

    y = y_c + y_p

  
  where y_c is the complementary function solving the associated homogeneous equation and y_p is any particular solution of the full equation.

II. Operator Method — constant-coefficient equations

A. Purpose and principle

The operator method represents differentiation by the symbol D, converting a differential equation into an algebraic equation in D. It is especially efficient when the coefficients are constants.

• Operator definition: Let
  

  TEXT

    D = d/dx

  
  Then Dy=y', D²y=y'', and generally Dⁿy=dⁿy/dxⁿ.

• Operator form: The equation
  

  TEXT

    aₙy⁽ⁿ⁾ + aₙ₋₁y⁽ⁿ⁻¹⁾ + ... + a₀y = f(x)

  
  becomes
  

  TEXT

    F(D)y = f(x)

  
  where F(D)=aₙDⁿ+...+a₀.

• Solution structure: Solve F(D)y=0 for y_c, then determine y_p from F(D)y_p=f(x).

B. Solution of non-homogeneous linear differential equations with constant coefficients using operator method

This method obtains the complementary function from an auxiliary equation and the particular integral by applying inverse operators.

• Auxiliary equation: Replace D by a number m in F(D)=0:
  

  TEXT

    F(m)=0

  
  Each root determines a complementary-function term.

• Root rules:

  • A real simple root m=a gives c e^{ax}.
  • A repeated root m=a of multiplicity r gives
    
    TEXT

        (c₁+c₂x+...+cᵣxʳ⁻¹)e^{ax}.

  • Complex roots a±ib give
    
    TEXT

        e^{ax}(c₁ cos bx+c₂ sin bx).

• Particular integral: Formally,
  

  TEXT

    y_p = [1/F(D)] f(x).

  
  The inverse operator is interpreted through algebraic identities, differentiation, and standard forms.

• Exponential forcing rule: For f(x)=e^{ax},
  

  TEXT

    1/F(D) e^{ax} = e^{ax}/F(a),

  
  provided F(a)≠0.

• Resonance: If F(a)=0, the preceding rule fails because e^{ax} already occurs in y_c; multiply the trial by the required power of x. For a root of multiplicity r, use xʳe^{ax} before evaluating the operator.

• Worked example: Solve
  

  TEXT

    (D²-3D+2)y=e^x.

  
  The auxiliary equation is (m-1)(m-2)=0, so y_c=c₁e^x+c₂e^{2x}. Since F(1)=0, resonance occurs and take y_p=Ax e^x. Substitution gives A=-1, hence
  

  TEXT

    y = c₁e^x+c₂e^{2x}-xe^x.

• Polynomial forcing: For f(x)=P_n(x), try a polynomial of degree n, unless resonance with the root m=0 requires multiplication by a power of x.

• Trigonometric forcing: For f(x)=a cos bx+b sin bx, use the complex-exponential representation or assume
  

  TEXT

    y_p=A cos bx+B sin bx.

  
  If ±ib are roots of F(m)=0, multiply the trial by the appropriate power of x.

III. Method of variation of parameters — variable coefficients and general forcing

A. Purpose and principle

Variation of parameters constructs a particular solution by replacing the constants in the complementary function with functions. Unlike the undetermined-coefficient method, it does not require the forcing term to have a special algebraic form.

• Standard equation: First write the second-order equation as
  

  TEXT

    y''+P(x)y'+Q(x)y=R(x).

• Complementary solutions: Let y₁ and y₂ be linearly independent solutions of
  

  TEXT

    y''+P(x)y'+Q(x)y=0.

  
  Their Wronskian is
  

  TEXT

    W(y₁,y₂)=y₁y₂'-y₂y₁'.

• Variable-constant trial: Instead of c₁y₁+c₂y₂, set
  

  TEXT

    y_p=u₁(x)y₁(x)+u₂(x)y₂(x).

B. Method of variation of parameters

The method determines u₁ and u₂ from two first-order equations, producing a particular solution through definite or indefinite integrals.

• Auxiliary condition: Impose
  

  TEXT

    u₁'y₁+u₂'y₂=0.

  
  This removes second derivatives of u₁ and u₂ when y_p is differentiated.

• Determining equations: Substitution into the differential equation gives
  

  TEXT

    u₁'y₁'+u₂'y₂'=R(x).

  
  Solving the two equations yields
  

  TEXT

    u₁'=-y₂R/W,
    u₂'= y₁R/W.

• Integral formulas: Therefore,
  

  TEXT

    u₁=-∫(y₂R/W)dx,
    u₂= ∫(y₁R/W)dx,

  
  and
  

  TEXT

    y_p=-y₁∫(y₂R/W)dx+y₂∫(y₁R/W)dx.

• Worked example: For
  

  TEXT

    y''+y=tan x,

  
  take y₁=cos x, y₂=sin x, and W=1. Thus
  

  TEXT

    u₁=-∫sin x tan x dx=-∫sin²x/cos x dx,
    u₂=∫cos x tan x dx=∫sin x dx=-cos x.

  
  Substitution into y_p=u₁cos x+u₂sin x gives a particular solution after simplifying the first integral.

• Linear independence condition: The Wronskian must be nonzero on the interval; otherwise y₁ and y₂ cannot serve as a fundamental pair.

• Role of constants: Constants arising from the integrals merely reproduce terms in y_c, so they may be set to zero when constructing one particular integral.

C. Applications and limitations

Variation of parameters is broadly applicable, but its integrals may be more difficult than the original equation.

• Broad applicability: It handles R(x) such as ln x, tan x, or arbitrary continuous functions, for which an undetermined trial is unavailable.

• Coefficient normalization: The formulas require the coefficient of y'' to be one; divide the equation by the leading coefficient first.

• Regularity requirement: Functions P, Q, and R should be continuous on the interval under consideration.

IV. Method of undetermined coefficient — trial-based particular solutions

A. Purpose and principle

The method of undetermined coefficients assumes a form for y_p based on the forcing function and determines unknown constants by substitution. It is mainly used for constant-coefficient equations with elementary forcing terms.

• Admissible forcing functions: Common forms include polynomials, exponentials, sines, cosines, and finite sums or products of these.

• Trial principle: If R(x)=e^{ax}P_n(x), use
  

  TEXT

    y_p=e^{ax}Q_n(x),

  
  where Q_n is a polynomial with unknown coefficients.

• Trigonometric trial: For R(x)=A cos bx+B sin bx, use
  

  TEXT

    y_p=C cos bx+D sin bx.

B. Method of undetermined coefficient

The coefficients in the trial are found by direct substitution and comparison of corresponding terms.

• Polynomial example: For
  

  TEXT

    y''-3y'+2y=x²,

  
  choose y_p=Ax²+Bx+C. Then
  

  TEXT

    y_p'=2Ax+B,  y_p''=2A.

  
  Substitution and comparison of coefficients of x², x, and constants determine A, B, and C.

• Sum rule: If the forcing is R=R₁+R₂, choose y_p=y_{p1}+y_{p2}. This follows from linearity and avoids solving all coefficients in one large expression.

• Product rule: For e^{ax}(P_n cos bx+Q_n sin bx), use
  

  TEXT

    y_p=e^{ax}[(A_nxⁿ+...+A₀)cos bx
              +(B_nxⁿ+...+B₀)sin bx].

• Resonance correction: If the trial duplicates a complementary-function term, multiply it by x. If duplication has multiplicity r, multiply by xʳ.

• Comparison with variation of parameters:

  1. Undetermined coefficients: Faster for standard forcing, but restricted to suitable elementary functions.
  2. Variation of parameters: More general, but requires evaluation of integrals and a known complementary pair.

• Coefficient comparison: After substitution, collect powers and functions such as x², x, 1, cos bx, and sin bx; equate their coefficients independently.

V. Solution of Euler-Cauchy equation — variable-power coefficients

A. Definition and standard form

An Euler–Cauchy equation has powers of the independent variable multiplying successive derivatives. Its coefficients are not constant, but the equation becomes constant-coefficient after the substitution x=e^t.

• Second-order form:
  

  TEXT

    ax²y''+bxy'+cy=f(x).

• Higher-order pattern: The coefficient of y⁽ʳ⁾ is proportional to xʳ, giving expressions such as x³y''', x²y'', and xy'.

• Domain assumption: Usually x>0 is taken so that t=ln x is real. Separate treatment may be needed across x=0.

B. Solution of Euler-Cauchy equation

For the homogeneous equation, a power trial gives an indicial equation; alternatively, the logarithmic substitution converts it into a constant-coefficient equation.

• Power trial: Assume
  

  TEXT

    y=x^m.

  
  Then y'=mx^{m-1} and y''=m(m-1)x^{m-2}. Substitution into
  

  TEXT

    ax²y''+bxy'+cy=0

  
  gives
  

  TEXT

    am(m-1)+bm+c=0,

  
  called the indicial or auxiliary equation.

• Root rules:

  • Distinct real roots m₁,m₂ give
    
    TEXT

        y_c=c₁x^{m₁}+c₂x^{m₂}.

  • Repeated root m gives
    
    TEXT

        y_c=x^m(c₁+c₂ln x).

  • Complex roots m=α±iβ give
    
    TEXT

        y_c=x^α[c₁cos(βln x)+c₂sin(βln x)].

• Logarithmic transformation: Put x=e^t, Y(t)=y(e^t). Then
  

  TEXT

    xy'=Y',
    x²y''=Y''-Y'.

  
  Thus the Euler–Cauchy equation becomes
  

  TEXT

    a(Y''-Y')+bY'+cY=f(e^t),

  
  or
  

  TEXT

    aY''+(b-a)Y'+cY=f(e^t),

  
  which can be solved by constant-coefficient methods.

• Non-homogeneous equations: After finding y_c, determine y_p using a power trial, operator techniques in t, or variation of parameters after transforming the equation.

• Worked example: Solve
  

  TEXT

    x²y''-3xy'+4y=0.

  
  With y=x^m, the indicial equation is
  

  TEXT

    m(m-1)-3m+4=m²-4m+4=(m-2)²=0.

  
  The repeated root is m=2, so
  

  TEXT

    y=x²(c₁+c₂ln x).

• Validity of solutions: The final solution is valid on an interval not crossing x=0; for x<0, expressions may be written using ln|x| where appropriate.

C. Applications and limitations

Euler–Cauchy equations model scale-invariant behavior, but their singular point requires careful domain selection.

• Scale structure: Replacing x by kx preserves the derivative-power pattern, which explains why powers x^m are natural trial solutions.

• Singular point: At x=0, coefficients such as x² vanish, so standard existence results may fail and initial conditions must be imposed away from zero.

• Method selection: Use the indicial equation for homogeneous problems; use x=e^t when the non-homogeneous term or higher order makes direct power trials inconvenient.