Solving Basic Algebraic Equations ax + y = z

Solving Algebraic Equations and

Other Symbolic Tools

Solving Basic Algebraic Equations ax + y = z

solve('a*x + y=z',‘x') x = -(y - z)/a

solve(equation, variable)

solve('a*x + y=z',‘a') a = -(y - z)/x

solve('a*x + y=z',‘y') y = z - a*x

Solving Quadratic Equations

𝐚 ∗ 𝐱^𝟐 + b*x + z = 0 𝐱^𝟐 - 6*x - 12 = 0 s = solve('x^2+ 6*x - 12 = 0')

s (1)= 3 + 21^(0.5) s (2)= 3 - 21^(0.5)

x1 = double (s(1)) = -7.5826 x1 = double (s(2)) = 1.5826

Plotting Symbolic Equations

𝐱^𝟐 + 6*x - 12 = 0 d = 'x^2 +6*x – 12';

ezplot(d)

-6 -4 -2 0 2 4 6

-20 -10 0 10 20 30 40 50 60 70

x x²+ 6 x - 12

Plotting Symbolic Equations

d = 'x^2 –6*x – 12';

ezplot(d,[-2,8])

ezplot(f, [x1 , x2 ])

-2 -1 0 1 2 3 4 5 6 7 8

-20 -15 -10 -5 0 5

x x²- 6 x - 12

Solving Higher Order Equations

(x+1)^2- (x-2) = 0 x^4–5*x^3+4*x^2–5*x+6 d= ‘x^4–5*x^3+4*x^2–5*x+6’

s=solve(d)

x1 = double (s(1)) = 4.2588 x2 = double (s(2)) = 1.1164

x3 = double (s(3)) = -0.1876 + 1.1076i x4 = double (s(4)) = -0.1876 - 1.1076i

-10 -8 -6 -4 -2 0 2 4 6 8 10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

x

x⁴-5 x³+4 x²-5 x+6

ezplot(d,[-10,10])

Systems of Equations

aX + bY = Z cX − dY = H

solve(equation1, equation2)

Example

x = s.x = 0.76 y = s.y = 0.21

s = solve('5*x + 4*y = 3','x – 6*y = 2'); Or s = solve('d1', 'd2');

5*x + 4*y = 3 x – 6*y = 2

Expanding and Collecting Equations

(x -1)* (x +4)

expand(sin(x+y)) ans = x^2 +3*x – 4

cos(x+y) syms x

expand((x–1)*(x+4)) Expanding

syms x syms y

ans = cos(x)*cos(y)–sin(x)*sin(y) (x + 2) (x − 3) = x2− x − 6

ans =(x–1)*(x+4)

factor(x^2 +3*x – 4) factor(cos(x)*cos(y)–sin(x)*sin(y))) ans = sin(x+y)

Expanding and Collecting Equations

x*(x^2− 2) = x^3− 2*x

simplify (x^3− 2*x)) ans = x^3− 2*x

syms x

collect (x*(x^2− 2)) Collecting

ans = x*(x^2− 2

Differential Equations

We can compute symbolic derivatives using MATLAB with a call to the diff command. Simply pass the function you want to differentiate to diff as this example shows:

syms x t f = x^2;

g = sin(10*t);

diff(f) ans = 2*x

diff(g) ans = 10*cos(10*t)

Higher derivatives

The second derivative of (t^3)

diff(f) syms t

f = t^3

f = 3*t^2 diff(f,2) f = 6*t Example

Plot the function f (x) = x4 –2x3 and show any local minima and maxima.

Solving ODE’s

𝐝𝐲

𝐝𝐱 = 𝟐 ∗ 𝐲 + 𝐜𝐨𝐬 𝐱

‘Dy = 2*y + cos(x)’

Example

𝒚^′′ + 𝒚^′ = 𝟓 ∗ 𝐬𝐢𝐧𝟕𝐱 'D2y + 2Dy = 5*sin(7*x)' dsolve(‘eqn’,‘cond1’, ‘cond2’,…)

eq1 ='Dy=a*y+cos(x)';

s=dsolve(eq1)

Ans [ s= -(cos(x) - C2*exp(a*t))/a]

C2 = 2; a = 4;

f = subs(s)

Ans [ f = exp(4*t)/2 - cos(x)/4]

𝐝𝐲

𝐝𝐱 = 𝐚 ∗ 𝐲 + 𝐜𝐨𝐬 𝐱 𝐂 = 𝟐 , 𝐚 = 𝟒

Solving ODE’s

Example

s =dsolve ('Dy = y*t/(t–5)','y(0) = 2') Ans [ s= -(2*exp(t + 5*log(t - 5)))/3125]

𝐝𝐲

𝐝𝐭 = 𝒕

𝒕 − 𝟓𝐲(𝐭) 𝐲(𝟎) = 2

Example; Second and higher order equations 𝒅^𝟐𝐲

𝒅𝒕^𝟐 − 𝐲 = 𝟎 𝐲(𝟎) = -1 𝒚^′(𝟎) = 2 s =dsolve('D2y – y = 0','y(0) = –1','Dy(0) = 2') Ans [ s= 1/2*exp(t) –3/2*exp(–t)]

Solving ODE’s

s = dsolve('DX = Y','DY = –X','X(0)= –1','Y(0)=2');

s.x

Ans [ –cos(t)+2*sin(t)]

s.Y

Ans [sin(t)+2*cos(t)]

𝐝𝐱

𝐝𝐭 = 𝐲 , ^𝐝𝐲

𝐝𝐭 = −𝐱 𝐲(𝟎) = 2 , x(0) = -1

Systems of Equations

Numerical Solution of ODEs

Integration 𝑥

dx =

(𝑛+1)

𝑥

int('x^n')

ans = x^(n+1)/(n+1) Example

What is the integral of f(x) = b^x ? Evaluate the resulting expression for b = 2, x = 4.

syms b x f = b^x;

F = int(f)

F = 1/log(b)*b^x subs(F,{b,x},{2,4}

ans = 23.0831

Integration

3𝑦 ∗ sec (𝑥)

int('3*y^2*sec(x))

int('3*y^2*sec(x)',y) ans = y^3*sec(x) int('3*y^2*sec(x)',x)

ans = 3*y^2*log(sec(x)+tan(x))

dx dy

Not acceptable

Definite Integration

𝐟 𝐱 𝐝𝐱 = 𝐟 𝐚 − 𝐟(𝐛)

𝐚

𝐛

𝐱 𝐝𝐱 = ^𝟏

𝟐 𝒙^𝟐 𝟑

𝟐 = ^𝟏

𝟐 ∗

𝟑

𝟐 𝟗 − ^𝟏

𝟐 ∗ 𝟒 = ^𝟓

𝟐

int('x',2,3) ans = 5/2

Integration

Example

What is the area under the curve f(x) = x^2cosx for −6 ≤ x ≤ 6?

syms x

f = x^2*cos(x);

s = int(f,–6,6)

s = 68*sin(6)+24*cos(6) double(s)

ans = 4.0438

Multidimensional Integration 𝑥𝑦

𝑧

dx dy dz

syms x y z

int (int (int(x*y^2*z^5,x),y),z) ans = 1/36*x^2*y^3*z^6

syms y x f = x^2*y;

F = int(int(f,x,2,4),y,1,2) F = 28

𝒙_𝟏^𝟐 _𝟐^{𝟒 𝟐}y dx dy

The Laplace Transform

syms t

laplace(t^5) ans =120/s^6

The Laplace transform of a function of time f(t) is given by

ℓ 𝐭

=

The Inverse Laplace Transform

syms s

ilaplace(120/s^6) ans =t^5

Solving Differential Equations

The Laplace transform simplifies differential equations, turning them into algebraic ones. The Laplace transform of the first derivative of a function is:

syms s

ilaplace(120/s^6) ans =t^5

And the Laplace transform of the second derivative of a function is: