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Troubleshoot Equation Solutions from solve Function

If solve returns solutions that look complicated, or if solve cannot handle an input, there are many options. These options simplify the solution space for solve. These options also help solve when the input is complicated, and might allow solve to return a solution where it was previously stuck.

Return Only Real Solutions

Solve the equation x^5 - 1 == 0. This equation has five solutions.

syms x
solve(x^5 - 1 == 0, x)
ans =
                                                     1
 - (2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 - 5^(1/2)/4 - 1/4
   (2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 - 5^(1/2)/4 - 1/4
   5^(1/2)/4 - (2^(1/2)*(5^(1/2) + 5)^(1/2)*1i)/4 - 1/4
   5^(1/2)/4 + (2^(1/2)*(5^(1/2) + 5)^(1/2)*1i)/4 - 1/4

If you only need real solutions, specify the Real option as true. The solve function returns the one real solution.

solve(x^5 - 1, x, 'Real', true)
ans =
1

Apply Simplification Rules

Solve the following equation. The solve function returns a complicated solution.

syms x
solve(x^(5/2) + 1/x^(5/2) == 1, x)
ans =
                                                       1/(1/2 - (3^(1/2)*1i)/2)^(2/5)
                                                       1/((3^(1/2)*1i)/2 + 1/2)^(2/5)
 -(5^(1/2)/4 - (2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 + 1/4)/(1/2 - (3^(1/2)*1i)/2)^(2/5)
 -((2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 + 5^(1/2)/4 + 1/4)/(1/2 - (3^(1/2)*1i)/2)^(2/5)
 -(5^(1/2)/4 - (2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 + 1/4)/(1/2 + (3^(1/2)*1i)/2)^(2/5)
 -((2^(1/2)*(5 - 5^(1/2))^(1/2)*1i)/4 + 5^(1/2)/4 + 1/4)/(1/2 + (3^(1/2)*1i)/2)^(2/5)

To apply simplification rules when solving equations, specify the IgnoreAnalyticConstraints option as true. The applied simplification rules are not generally correct mathematically but might produce useful solutions, especially in physics and engineering. With this option, the solver does not guarantee the correctness and completeness of the result.

solve(x^(5/2) + 1/x^(5/2) == 1, x, 'IgnoreAnalyticConstraints', true)
ans =
 1/(1/2 - (3^(1/2)*1i)/2)^(2/5)
 1/((3^(1/2)*1i)/2 + 1/2)^(2/5)

This solution is simpler and more usable.

Use Assumptions to Narrow Results

For solutions to specific cases, set assumptions to return appropriate solutions. Solve the following equation. The solve function returns seven solutions.

syms x
solve(x^7 + 2*x^6 - 59*x^5 - 106*x^4 + 478*x^3 + 284*x^2 - 1400*x + 800, x)
ans =
                  1
      - 5^(1/2) - 1
 - 17^(1/2)/2 - 1/2
   17^(1/2)/2 - 1/2
         -5*2^(1/2)
          5*2^(1/2)
        5^(1/2) - 1

Assume x is a positive number and solve the equation again. The solve function only returns the four positive solutions.

assume(x > 0)
solve(x^7 + 2*x^6 - 59*x^5 - 106*x^4 + 478*x^3 + 284*x^2 - 1400*x + 800, x)
ans =
                1
 17^(1/2)/2 - 1/2
        5*2^(1/2)
      5^(1/2) - 1

Place the additional assumption that x is an integer using in(x,'integer'). Place additional assumptions on variables using assumeAlso.

assumeAlso(in(x,'integer'))
solve(x^7 + 2*x^6 - 59*x^5 - 106*x^4 + 478*x^3 + 284*x^2 - 1400*x + 800, x)
ans =
1

solve returns the only positive, integer solution to x.

Clear the assumptions on x for further computations by recreating it using syms.

syms x

Alternatively, to make several assumptions, use the & operator. Make the following assumptions, and solve the following equations.

syms a b c f g h y
assume(f == c & a == h & a~= 0)
S = solve([a*x + b*y == c, h*x - g*y == f], [x, y], 'ReturnConditions', true);
S.x
S.y
S.conditions
ans =
f/h
ans =
0
ans =
b + g ~= 0

Under the specified assumptions, the solution is x = f/h and y = 0 under the condition b + g ~= 0.

Clear the assumptions on the variables for further computations by recreating them using syms.

syms a c f h

Simplify Solutions

The solve function does not call simplification functions for the final results. To simplify the solutions, call simplify.

Solve the following equation. Convert the numbers to symbolic numbers using sym to return a symbolic result.

syms x
S = solve((sin(x) - 2*cos(x))/(sin(x) + 2*cos(x)) == 1/2, x)
S =
 -log(-(- 140/37 + 48i/37)^(1/2)/2)*1i
  -log((- 140/37 + 48i/37)^(1/2)/2)*1i

Call simplify to simplify solution S.

simplify(S)
ans =
        -log(37^(1/2)*(- 1/37 - 6i/37))*1i
 log(2)*1i - (log(- 140/37 + 48i/37)*1i)/2

Call simplify with more steps to simplify the result even further.

simplify(S, 'Steps', 50)
ans =
 atan(6) - pi
      atan(6)

Tips

  • To represent a number exactly, use sym to convert the number to a floating-point object. For example, use sym(13)/5 instead of 13/5. This represents 13/5 exactly instead of converting 13/5 to a floating-point number. For a large number, place the number in quotes. Compare sym(13)/5, sym(133333333333333333333)/5, and sym('133333333333333333333')/5.

    sym(13)/5
    sym(133333333333333333333)/5
    sym('133333333333333333333')/5
    ans =
    13/5
    ans =
    133333333333333327872/5
    ans =
    133333333333333333333/5

    Placing the number in quotes and using sym provides the highest accuracy.

  • If possible, simplify the system of equations manually before using solve. Try to reduce the number of equations, parameters, and variables.