Improve JointEq error message for second-order ODEs and add documentation (#798)

* Initial plan

* Improve JointEq error message and add tests for second-order ODEs

Co-authored-by: Routhleck <88108241+Routhleck@users.noreply.github.com>

* Add documentation for second-order ODEs with JointEq

Co-authored-by: Routhleck <88108241+Routhleck@users.noreply.github.com>

* Fix trailing whitespace in test_joint_eq.py

Co-authored-by: Routhleck <88108241+Routhleck@users.noreply.github.com>

* Update joint equations tutorial outputs and metadata

Refreshed code cell outputs, execution counts, and execution metadata in the joint_equations.ipynb tutorial. Added print outputs for the second-order system and Jansen-Rit model examples, and updated function signatures and object references to reflect current execution results.

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: Routhleck <88108241+Routhleck@users.noreply.github.com>
Co-authored-by: routhleck <sichaohe@stu.pku.edu.cn>

Author: Copilot

brainpy/integrators/joint_eq.py

@@ -154,8 +154,18 @@ def __init__(self, *eqs):
             vars, _, _ = _get_args(eq)
             for var in vars:
                 if var in vars_in_eqs:
-                    raise DiffEqError(f'Variable "{var}" has been used, however we got a same '
-                                      f'variable name in {eq}. Please change another name.')
+                    raise DiffEqError(
+                        f'Variable "{var}" has been used, however we got a same '
+                        f'variable name in {eq}.\n\n'
+                        f'In JointEq, each state variable should appear as the first parameter '
+                        f'before "t" in exactly one derivative function. If "{var}" is a state '
+                        f'variable in another equation, it should be placed AFTER "t" in this '
+                        f'function as a dependency.\n\n'
+                        f'Correct signature pattern:\n'
+                        f'  def d{var}({var}, t, <dependencies>): ...  # {var} is the state variable\n'
+                        f'  def dOther(other, t, {var}): ...           # {var} is a dependency\n\n'
+                        f'Current function signature: {inspect.signature(eq)}'
+                    )
             vars_in_eqs.extend(vars)
             self.vars_in_eqs.append(vars)
 

brainpy/integrators/tests/test_joint_eq.py

@@ -128,3 +128,46 @@ def test_nested_joint_eq1(self):
         EQ2 = JointEq(EQ1, dn)
         EQ3 = JointEq(EQ2, dV)
         print(EQ3(m=0.1, h=0.2, n=0.3, V=10., t=0., I=0.))
+
+    def test_second_order_ode(self):
+        """Test second-order ODE system (e.g., harmonic oscillator)"""
+        # Second-order ODE: d²x/dt² = -k*x - c*dx/dt
+        # Split into: dx/dt = v, dv/dt = -k*x - c*v
+        k = 1.0  # spring constant
+        c = 0.1  # damping
+
+        def dx(x, t, v):
+            """dx/dt = v"""
+            return v
+
+        def dv(v, t, x):
+            """dv/dt = -k*x - c*v"""
+            return -k * x - c * v
+
+        # Create joint equation
+        eq = JointEq(dx, dv)
+
+        # Test call
+        result = eq(x=1.0, v=0.0, t=0.0)
+        self.assertEqual(len(result), 2)
+        self.assertEqual(result[0], 0.0)  # dx/dt = v = 0
+        self.assertEqual(result[1], -k * 1.0)  # dv/dt = -k*x
+
+    def test_second_order_ode_wrong_signature(self):
+        """Test that wrong signature gives helpful error message"""
+        # WRONG: both x and v before t in dx function
+        def dx_wrong(x, v, t):
+            return v
+
+        def dv(v, t, x):
+            return -x
+
+        # This should raise an error with helpful message
+        with self.assertRaises(DiffEqError) as cm:
+            JointEq(dx_wrong, dv)
+
+        # Check that error message is helpful
+        error_msg = str(cm.exception)
+        self.assertIn('state variable', error_msg.lower())
+        self.assertIn('AFTER "t"', error_msg)
+        self.assertIn('dependency', error_msg.lower())