feat: enhance Gradio interfaces for first-order and second-order ODE solvers with improved descriptions and examples

maths/differential_equations/solve_first_order_ode.py

@@ -67,7 +67,7 @@ first_order_ode_interface = gr.Interface(
     fn=lambda ode_str, t_span_str, y0_str, t_eval_count, method: solve_first_order_ode(
         string_to_ode_func(ode_str, ('t', 'y')),
         parse_time_span(t_span_str),
-        parse_float_list(y0_str), # y0 can be a list for systems
+        parse_float_list(y0_str),
         int(t_eval_count),
         method
     ),
@@ -87,9 +87,20 @@ first_order_ode_interface = gr.Interface(
         gr.JSON(label="Raw Data (t, y values)", visible=lambda res: res['success']) # For users to copy if needed
     ],
     title="First-Order ODE Solver",
-    description="Solves dy/dt = f(t,y) or a system of first-order ODEs. " \
-                "WARNING: Uses eval() for the ODE function string - potential security risk. " \
-                "For systems, `y` in lambda is `[y1, y2, ...]`, return `[dy1/dt, dy2/dt, ...]`. " \
-                "Example (Damped Oscillator): ODE: lambda t, y: [y[1], -0.5*y[1] - y[0]], y0: 1,0, Timespan: 0,20",
+    description="""
+Solves dy/dt = f(t, y) or a system of first-order ODEs.
+
+- Enter a Python lambda for the ODE (e.g., `lambda t, y: -y*t`).
+- For systems, use `y` as a list: `lambda t, y: [y[1], -0.1*y[1] - y[0]]`.
+- Initial conditions: single value (e.g., `1`) or comma-separated for systems (e.g., `1,0`).
+
+**Examples:**
+
+- Simple: `lambda t, y: -y*t`, y0: `1`, t_span: `0,5`
+- System: `lambda t, y: [y[1], -0.1*y[1] - y[0]]`, y0: `1,0`, t_span: `0,20`
+- Lotka-Volterra: `lambda t, y: [1.5*y[0] - 0.8*y[0]*y[1], 0.5*y[0]*y[1] - 0.9*y[1]]`, y0: `10,5`, t_span: `0,20`
+
+WARNING: Uses eval() for the ODE function string - potential security risk.
+""",
     flagging_mode="manual"
 )

maths/differential_equations/solve_second_order_ode.py

@@ -104,8 +104,18 @@ second_order_ode_interface = gr.Interface(
         gr.JSON(label="Raw Data (t, y, dy_dt values)", visible=lambda res: res['success'])
     ],
     title="Second-Order ODE Solver",
-    description="Solves d²y/dt² = f(t, y, dy/dt). "
-                "WARNING: Uses eval() for the ODE function string - potential security risk. "
-                "Example (Pendulum): ODE: lambda t, y, dy_dt: -9.81/1.0 * math.sin(y), y0: math.pi/4, dy0/dt: 0, Timespan: 0,10",
+    description="""
+        Solves d²y/dt² = f(t, y, dy/dt) for a single second-order ODE.
+
+        - Enter a Python lambda for the ODE (e.g., `lambda t, y, dy_dt: -0.1*dy_dt - math.sin(y)`).
+        - Initial y(t_start) and dy/dt(t_start) must be single values.
+
+        **Examples:**
+
+        - Damped oscillator: `lambda t, y, dy_dt: -0.1*dy_dt - y`, y0: `1.0`, dy0/dt: `0.0`, t_span: `0,20`
+        - Simple pendulum: `lambda t, y, dy_dt: -9.81/1.0 * math.sin(y)`, y0: `math.pi/4`, dy0/dt: `0`, t_span: `0,10`
+
+        WARNING: Uses eval() for the ODE function string - potential security risk.
+        """,
     flagging_mode="manual"
 )