Differential equations first look

A differential equation relates a function to its derivatives. An ordinary differential equation has one independent variable, often thought of as time. The unknown is a function, not a number. Growth, cooling, and simple oscillators are written this way because the sentence you can say is about a rate. This pillar is a first look: what a solution is, how separation works, and how a slope field tells you the story before a formula appears. The long walkthrough of separation already lives in an explainer. This page is the map around it.

Who this pillar is for

Start here if a first course in differential equations is beginning, or if you can separate variables by ritual and lose constant solutions without noticing. You might be coming from an AP Calculus AB course that stopped at a brief introduction, or from calculus foundations once derivatives and integrals have meanings.

The AP Calculus AB study pathway reaches this topic as a late step, after series. Use that pathway if you want the whole calculus route. Use this page if the equation itself is the question. You do not need a full series course to understand separation. You do need to know what an integral accumulates.

Prerequisites

You need derivatives as instantaneous rates, the chain rule, and integrals of common functions. Implicit differentiation helps when you check a solution that is not written as $y$ equal to a formula. If any of that is shaky, read what a derivative actually measures and the integral section of calculus foundations.

The short definitions are differential equation, initial value problem, and slope field. OpenStax Calculus and MIT OpenCourseWare (including the differential equations course often numbered 18.03) are on the sources page. This pillar stops where the explainer on this site stops: separation, equilibria, and the habit of checking. Later methods are named so you know what a book is for.

What you should be able to do

When you finish the core, you should be able to do these things on a first order equation $y'=f(t,y)$.

• Say what a solution is: a function whose slope matches the rule at each point on its graph.
• Use an initial condition to choose one curve from a family, and name that pair an initial value problem.
• Separate $y'=g(t)h(y)$ when the form allows it, and test constant solutions before you divide.
• Sketch the idea of a slope field: short segments whose slopes are $f(t,y)$, with solution curves threading them.
• Find equilibria where $y'=0$, and say whether nearby solutions move toward them or away.
• Audit an explicit formula by differentiating it. A candidate that does not reproduce the equation is not a solution.

What a solution means

Solving $y'=f(t,y)$ means finding $y(t)$ whose derivative equals $f(t, y(t))$ on some interval. The graph has to follow the slopes the rule assigns. An initial condition $y(t_0)=y_0$ selects one curve from a family. Without it, a constant is still waiting to be chosen.

A common theorem says that if $f$ and its partial derivative with respect to $y$ are continuous near the start, then a unique solution exists on some interval around that start. The precise statement is in a text on the sources page. When the smoothness fails, two curves can leave the same point.

Separable equations

If $y'=g(t)h(y)$, the rate factors into a piece in $t$ and a piece in $y$. You can often integrate both sides after rearranging. That is the workhorse of a first look. The full write up is the separable walkthrough. The checkpoint below is the one to do cold.

Consider $dy/dx = 2xy$. The right hand side is zero whenever $y=0$, so the constant function $y=0$ is a solution. For other solutions, divide by $y$ and integrate:

$$\int \frac{1}{y}\,dy = \int 2x\,dx.$$

You get $\ln|y| = x^{2}+C$, hence $y = Ae^{x^{2}}$ with $A$ a nonzero constant. The zero solution is the case the division hid. Some people write $A=0$ afterward to include it. That is harmless only if you noticed the division. Differentiate a nonzero answer: if $y=Ae^{x^{2}}$, then $y'=2xAe^{x^{2}}=2xy$, which matches. An initial condition $y(0)=3$ forces $A=3$. That pair is an initial value problem.

The Riemann sum calculator is a picture of accumulation, not a symbolic integrator. The numerical derivative calculator can audit a slope after you differentiate by hand. For $y=3e^{x^{2}}$ at $x=0$, the slope should be $0$, because $y'=2xy$ and $x=0$.

Slope fields and equilibria

Before solving, sketch a slope field: at sample points, a short segment of slope $f(t,y)$. A solution has to thread those segments. This site has no slope field tool. A dozen segments on paper are enough.

Equilibria are constant solutions, where an autonomous rule $y'=f(y)$ is zero. Nearby solutions may approach that height or leave it. That question is often more useful than a formula.

Take the autonomous equation $y'=y(1-y)$. The equilibria are $y=0$ and $y=1$. Between them the slope is positive, so solutions increase toward $1$. Above $1$ the slope is negative, so solutions decrease toward $1$. Below $0$ the slope is negative, so solutions decrease and move away from $0$. In that rough portrait, $y=1$ attracts from above and from the interval between the equilibria, and $y=0$ does not attract. You can say this without solving. The separable explainer treats a similar autonomous equation and shows the partial fractions that produce a formula. Those coefficients are on the partial fraction decomposition calculator if you want the algebra written out. Do the portrait first.

What a first course adds next

Not every equation separates. A linear equation $y'+P(x)y=Q(x)$ is the usual next family. An integrating factor is a function of $x$ chosen so the left side becomes the derivative of a product. The integrating factor calculator writes $\mu(x)=e^{\int P\,dx}$, multiplies through, and integrates when that integral is elementary. When it is not, the page labels a Runge-Kutta table instead of inventing a formula. The books on the sources page write the same method. Separation was never the whole subject.

Constant-coefficient second order equations are another standard chapter. For $a y''+b y'+c y=0$, the characteristic equation $a r^2+b r+c=0$ decides whether the solution is two real exponentials, a repeated root with an extra factor of $x$, or a complex pair written with sine and cosine. The second order differential equation calculator writes the discriminant, the three cases, and the constants from $y(x_0)$ and $y'(x_0)$. Those roots are an ordinary quadratic. The quadratic formula calculator reports the discriminant on its own, without the differential equation wrapped around it. The second order page does not replace the separable checkpoint above.

A set of solutions is useful as a fundamental set only when the solutions are linearly independent. The Wronskian is the determinant of the matrix whose rows are the functions and their derivatives. For solutions of one linear homogeneous equation, Abel's identity says that determinant is either always zero or never zero, so one sample settles the question. For arbitrary formulas, a zero at one point does not prove dependence. The Wronskian calculator writes the matrix, the determinant, and that caveat.

The Laplace transform is the usual next tool once initial conditions are part of the question. It turns a constant-coefficient equation into algebra for a function of s, and the inverse step is partial fractions plus a short table. The Laplace transform calculator names the table row, the shift, and the inverse. It does not replace the separation checkpoint above, and it does not build the transformed equation for you.

A later linear system can be written with a matrix, and the eigenvalues describe growth along special directions. That door is linear algebra and eigenvalues in plain language. It is not required for the separable checkpoint. Several independent variables wait for the multivariable pillar.

Suggested learning order

Work in this order. Check every formula by differentiation before you trust the algebra that produced it.

1. Read derivative, definite integral, differential equation, initial value problem, and slope field.
2. If the rate sentence is empty, read what a derivative actually measures and the relevant parts of calculus foundations.
3. Separate $dy/dx=2xy$ on paper, including $y=0$. Impose $y(0)=3$ and differentiate the result.
4. Read the separable walkthrough and compare its warnings with the solution you just produced.
5. Sketch a slope field for $y'=y(1-y)$ at a few heights. Mark the equilibria before you draw curves.
6. Use the numerical derivative calculator only to audit an explicit solution. Use the Riemann sum calculator only if the integral itself is the fog.
7. If you want the calculus route that arrives here on purpose, follow the AP Calculus AB study pathway. For $y'+P(x)y=Q(x)$, use the integrating factor calculator. For systems, use the sources page.

Checkpoints and practice habits

Separation check. $y=3e^{x^{2}}$ should satisfy $y'=2xy$ and $y(0)=3$. At $x=0$ the slope is $0$. At $x=1$ the slope is $6e$. The numerical derivative calculator can estimate that slope. Your derivative is the authority. The estimate is a witness.

Equilibrium check. For $y'=y(1-y)$, name the two constant solutions and say which way a solution moves if it starts at $y=1/2$ and if it starts at $y=-1/2$. Increasing toward $1$, and decreasing away from $0$, are the answers.

Lost solution check. Separate $y'=y$ and also test $y=0$. The exponential family $y=Ae^{t}$ includes the zero solution when $A=0$. Say so on purpose. The separable explainer's first example is this equation with an initial value.

Common pitfalls

Existence on a small interval is not existence for all time. A solution can blow up at a finite time even when the rule looks smooth at the start. If a formula has a denominator that hits zero, ask whether the solution only lived up to that time.

Related reading

The computational home of this topic on the site is the separable walkthrough. The calculus cluster around it is calculus foundations and the AP Calculus AB study pathway. Eigenvalues for later linear systems live on linear algebra and in eigenvalues in plain language. Several independent variables wait for multivariable and vector calculus. Compare routes on the pathways hub.