Method of Variation of Parameters to Solve 2nd Order Differential Equations in MATLAB

MATLAB can be used to solve numerically second and higher-order ordinary differential equations. In this article, we will see the method of variation of parameters to Solve 2^nd Order Differential Equations in MATLAB.

Step 1: Let the given 2nd Order Differential Equation in terms of 'x' is:

\frac{d^{2}y}{dx^{2}} + p \frac{dy}{dx} + qy = f(x)

Step 2: Then, we reduce it to its Auxiliary Equation(AE) form: _{r}{^2}+ pr +Q = 0

Step 3: Then, we find the Determinant of the above AE by the Relation: Det  = ^ { p^2 - 4Q}

Step 4: If the Determinant found above is Positive (2 Distinct Real roots r₁ & r₂), then the Complementary Function(CF) will be:y = C1{e^{r1^x} + e^{r2^x}}

Step 5: If the Determinant found above is Zero (1 Unique Real Root ,r₁=r₂=r), then the Complementary Function(CF) will be: y = C1{e^{rx} + C2 X e^{rx}}

Step 6: If the Determinant found above is Negative (Complex Roots, r = α ± iβ), then the Complementary Function(CF) will be: y = {e^{ax} (C1 cos{\beta X })+ i (C2 Sin X {\beta x}})

Step 7: In all the Above 3-Cases, the Coefficient of C₁ is termed 'y₁', and the Coefficient of C₂ is termed 'y₂'_.

Step 8: Then we find the Wronskian(W) by the Relation: W(y_{1}, y_{2}) = y_{1}({y_{2}')- {y_{2}y_{1}'

Step 9: After finding 'W', we find the Particular Integral (PI) by the Relation: PI = -y_{1}\int_{}^{}\frac{y{2}f(x)}{w}dx + y_{2}\int_{}^{}  \frac{y{1}f(x)}{w} dx

Step 10: Finally the General Solution(GS) of the 2^nd Order Differential Equation is found by the Relation:  GS = CF + PI

Example:

% MATLAB code for Method of Variation of Parameters  % to Solve 2nd Order Differential  % Equations in MATLAB clear all  clc  % To Declare them as Variables syms r c1 c2 x  disp("Method of Variation of Parameters to Solve  2nd Order Differential Equations in MATLAB | GeeksforGeeks") E=input("Enter the coefficients of the 2nd Order Differential equation");  X=input("Enter the R.H.S of the 2nd order Differential equation"); % Coefficients of the 2nd Order Differential Equations AE=a*r^2+b*r+c; a=E(1); b=E(2); c=E(3);  S=solve(AE); % Roots of Auxiliary Equation (AE) r1=S(1); r2=S(2); % Determinant of Auxiliary Equation (AE) D=b^2-4*a*c;  if D>0  y1=exp(r1*x);   y2=exp(r2*x);  % Complementary Function  cf=c1*y1+c2*y2  elseif D==0  y1=exp(r1*x);  y2=x*exp(r2*x);   % Complementary Function  cf=c1*y1+c2*y2  else  alpha=real(r1);   beta=imag(r2);  y1=exp(alpha*x)*cos(beta*x);   y2=exp(alpha*x)*sin(beta*x);  % Complementary Function  cf=c1*y1+c2*y2 end W=simplify(y1*diff(y2,x)-y2*diff(y1,x));  % Particular Integral PI=simplify((-y1)*int(y2*X/W) + (y2)*int(y1*X/W))  % General Solution GS=simplify(cf+PI)  

Output:

Input: y'' -6y' + 25 = e2x + sinx + x

Input: y'' -2y' + 3 = x3 + cosx