Calculator Inputs
Select the equation family that best matches your problem.
Constants C₁ and C₂ are used to draw one sample curve.
Solution Graph
The graph shows one representative curve using your chosen arbitrary constants.
Example Data Table
| Model |
Sample equation |
Inputs |
General solution form |
| First order linear |
y' + 2y = 5 |
P = 2, Q = 5 |
y = C₁e-2x + 2.5 |
| Separable power |
y' = 3xy² |
k = 3, m = 1, n = 2 |
y = [(-1)(1.5x² + C₁)]-1 |
| Second order homogeneous |
y'' - 3y' + 2y = 0 |
a = 1, b = -3, c = 2 |
y = C₁e2x + C₂ex |
| Exact form |
(2x + y)dx + (x + 3y)dy = 0 |
A = 2, B = 1, C = 1, D = 3 |
x² + xy + 1.5y² = C₁ |
Formula Used
First order linear
For y' + Py = Q, use μ(x)=ePx.
The solution is y = C₁e-Px + Q/P when P is not zero.
Bernoulli equation
For y' + Py = Qyⁿ, set z = y1-n.
Then solve z' + (1-n)Pz = (1-n)Q.
Separable power
For y' = kxᵐyⁿ, separate y-ndy = kxᵐdx.
Integrate each side and solve for y when possible.
Second order
For ay'' + by' + cy = 0, solve ar² + br + c = 0.
Root type controls the solution form.
Constant forcing
For ay'' + by' + cy = d, add a particular solution to the complementary solution.
Exact equation
For Mdx + Ndy = 0, exactness requires ∂M/∂y = ∂N/∂x.
Then F(x,y)=C₁.
How to Use This Calculator
- Select the equation type that matches your problem.
- Enter only the coefficients needed for that selected model.
- Set C₁ and C₂ to draw a representative solution curve.
- Choose the x range and number of plot points.
- Press the calculate button and read the result panel.
- Download the CSV or PDF file for study records.
About This Calculator
A general solution is the broad answer to a differential equation. It contains arbitrary constants. Those constants represent a whole family of curves. This calculator helps you study that family before applying initial values. It supports several common classroom models. You can test first order linear equations. You can explore Bernoulli equations. You can check separable power equations. You can solve second order equations with constant coefficients. You can also inspect simple exact equations.
Why General Solutions Matter
Differential equations describe change. They appear in motion, growth, circuits, heat flow, finance, and control systems. A general solution keeps the full structure visible. It shows the natural response. It also shows the forced response when a constant input exists. This is useful before fitting a real measurement. It is also useful during exam practice. You can see how roots change the final form. Real distinct roots create two exponential modes. Repeated roots add an extra x factor. Complex roots create oscillation with sine and cosine terms.
How The Tool Helps
The calculator separates the equation type from the coefficients. That makes each method easier to review. The result panel shows the interpreted equation. It then gives the solution form. It also explains the method used. The graph uses chosen constants. So it displays one member of the solution family. Change the constants to compare nearby curves. Use the range fields to zoom in or out. Use more plot points for smoother curves. Use fewer points for a quick check.
Best Study Workflow
Start with an equation from your notes. Select the matching model. Enter the coefficients carefully. Keep the default constants first. Press calculate. Read the method line. Then compare it with your manual work. Next, change the arbitrary constants. Watch the graph move. This builds intuition about solution families. Export the result when you need a record. Use the CSV file for spreadsheets. Use the PDF file for notes or reports. The tool is not a symbolic algebra engine for every possible equation. It is a guided solver for important forms. Always verify special cases and domain limits. This habit improves accuracy and deeper long term recall.
FAQs
1. What is a general solution?
A general solution includes arbitrary constants. It represents a family of solutions, not one single curve. Initial conditions later choose a specific member from that family.
2. Does this solve every differential equation?
No. It supports selected common forms. These include linear, Bernoulli, separable power, constant-coefficient second order, constant forcing, and simple exact equations.
3. Why do I enter C₁ and C₂?
The constants define one sample curve for the graph. The displayed formula still shows the general family with arbitrary constants.
4. What happens when roots are complex?
Complex roots create sine and cosine terms. The graph may show oscillation, decay, growth, or a mix, depending on the real part.
5. What is an exact equation?
An exact equation has a potential function F(x,y). Its differential matches Mdx + Ndy. The solution is written as F(x,y)=C₁.
6. Why is the graph sometimes blank?
Some formulas have domain limits. A blank graph can occur when powers, logarithms, or implicit branches do not produce real finite values in the selected range.
7. Can I use decimals?
Yes. All coefficient fields accept decimal values. Scientific notation may also work if your browser allows it in number fields.
8. Is the exported CSV based on the graph?
Yes. The CSV contains plotted x and y values for the representative curve. It also includes the equation and solution summary.