Solving Equations

Equations use ==. Solutions come back as lists of rules.

Solve

Solve[x^2 - 5 x + 6 == 0, x]
Solve[x^2 == 2, x]
Solve[a x + b == 0, x]
Solve[{x + y == 10, x - y == 2}, {x, y}]

{{x -> 2}, {x -> 3}}
{{x -> -Sqrt[2]}, {x -> Sqrt[2]}}
{{x -> -(b/a)}}
{{x -> 6, y -> 4}}

• Each inner list is one solution.
• Solutions are exact, and symbolic parameters like a and b are fine.

Choosing the domain

Solve[x^2 + 1 == 0, x]
Solve[x^2 + 1 == 0, x, Reals]
Reduce[x^2 < 4, x]

{{x -> -I}, {x -> I}}
{}
-2 < x < 2

• By default, Solve finds complex solutions. Add Reals or Integers to restrict them — {} means "no solutions".
• Reduce handles inequalities and describes all solutions, including special cases.

Numerical solutions

NSolve[x^3 - 2 x - 5 == 0, x, Reals]
FindRoot[Cos[x] == x, {x, 1}]

{{x -> 2.09455}}
{x -> 0.739085}

• NSolve finds all numeric solutions of polynomial equations.
• FindRoot searches near a starting point (x = 1 here) and works for any equation — but finds only one root.

Differential equations

DSolve[y'[x] == y[x], y[x], x]
DSolve[{y'[x] == y[x], y[0] == 2}, y[x], x]

{{y[x] -> E^x C[1]}}
{{y[x] -> 2 E^x}}

C[1] is an arbitrary constant (notebooks display it as a subscripted c₁). An initial condition fixes it.

For equations without a formula solution, NDSolve returns an interpolating function you can evaluate and plot:

sol = NDSolve[{y'[x] == -y[x], y[0] == 1}, y, {x, 0, 5}];
y[1] /. sol

{0.367879}

That's e⁻¹, as expected. Plot[y[x] /. sol, {x, 0, 5}] draws the solution.

Common mistakes

• Solve[x^2 = 4, x] — a single = assigns 4 to x^2 and fails with a message. Equations need ==.
• Using a variable that already has a value: if x = 3, then Solve[x^2 == 4, x] complains that 3 is not a valid variable. Clear[x] first.

Practice

Find where the line y = 2 x + 1 meets the parabola y = x^2 - 2.

Solve[{y == 2 x + 1, y == x^2 - 2}, {x, y}]

{{x -> -1, y -> -1}, {x -> 3, y -> 7}}