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 2nd Order Differential Equations in MATLAB.
Step 1: Let the given 2nd Order Differential Equation in terms of ‘x’ is:
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:
Step 4: If the Determinant found above is Positive (2 Distinct Real roots r1 & r2), then the Complementary Function(CF) will be:
Step 5: If the Determinant found above is Zero (1 Unique Real Root ,r1=r2=r), then the Complementary Function(CF) will be:
Step 6: If the Determinant found above is Negative (Complex Roots, r = α ± iβ), then the Complementary Function(CF) will be:
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:
Step 9: After finding ‘W’, we find the Particular Integral (PI) by the Relation:
Step 10: Finally the General Solution(GS) of the 2nd Order Differential Equation is found by the Relation:
Example:
Matlab
% 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 | w3wiki") 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