Solve the first-order differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}.$

Specify the first-order derivative by using diff and the equation by using ==. Then, solve the equation by using dsolve.

syms y(t) a
eqn = diff(y,t) == a*y;
S = dsolve(eqn)

S = $$C_1\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}t}$$

The solution includes a constant. To eliminate constants, see Solve Differential Equations with Conditions. For a full workflow, see Solving Partial Differential Equations.

Solve the second-order differential equation $\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dt}}^2}=\mathit{ay}.$

Specify the second-order derivative of y by using diff(y,t,2) and the equation by using ==. Then, solve the equation by using dsolve.

syms y(t) a
eqn = diff(y,t,2) == a*y;
ySol(t) = dsolve(eqn)

ySol(t) = $$C_1\hspace{0.17em}{\mathrm{e}}^{-\sqrt{a}\hspace{0.17em}t}+C_2\hspace{0.17em}{\mathrm{e}}^{\sqrt{a}\hspace{0.17em}t}$$

Solve the first-order differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}$ with the initial condition $$y(0)=5$$.

Specify the initial condition as the second input to dsolve by using the == operator. Specifying condition eliminates arbitrary constants, such as C1, C2, ..., from the solution.

syms y(t) a
eqn = diff(y,t) == a*y;
cond = y(0) == 5;
ySol(t) = dsolve(eqn,cond)

ySol(t) = $$5\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}t}$$

Next, solve the second-order differential equation $\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dt}}^2}={\mathit{a}}^2\mathit{y}$ with the initial conditions $$y(0)=b$$ and $$y^{\prime }(0)=1$$.

Specify the second initial condition by assigning diff(y,t) to Dy and then using Dy(0) == 1.

syms y(t) a b
eqn = diff(y,t,2) == a^2*y;
Dy = diff(y,t);
cond = [y(0)==b, Dy(0)==1];
ySol(t) = dsolve(eqn,cond)

ySol(t) = 
$$\frac{{\mathrm{e}}^{a\hspace{0.17em}t}\hspace{0.17em}\left(a\hspace{0.17em}b+1\right)}{2\hspace{0.17em}a}+\frac{{\mathrm{e}}^{-a\hspace{0.17em}t}\hspace{0.17em}\left(a\hspace{0.17em}b-1\right)}{2\hspace{0.17em}a}$$

This second-order differential equation has two specified conditions, so constants are eliminated from the solution. In general, to eliminate constants from the solution, the number of conditions must equal the order of the equation.

Solve the system of differential equations

Specify the system of equations as a vector. dsolve returns a structure containing the solutions.

syms y(t) z(t)
eqns = [diff(y,t) == z, diff(z,t) == -y];
S = dsolve(eqns)

S = struct with fields:
    z: [1x1 sym]
    y: [1x1 sym]

Access the solutions by addressing the elements of the structure.

ySol(t) = S.y

ySol(t) = $$C_1\hspace{0.17em}\cos \left(t\right)+C_2\hspace{0.17em}\sin \left(t\right)$$

zSol(t) = S.z

zSol(t) = $$C_2\hspace{0.17em}\cos \left(t\right)-C_1\hspace{0.17em}\sin \left(t\right)$$

When solving for multiple functions, dsolve returns a structure by default. Alternatively, you can assign solutions to functions or variables directly by explicitly specifying the outputs as a vector. dsolve sorts the outputs in alphabetical order using symvar.

Solve a system of differential equations and assign the outputs to functions.

syms y(t) z(t)
eqns = [diff(y,t)==z, diff(z,t)==-y];
[ySol(t),zSol(t)] = dsolve(eqns)

ySol(t) = $$C_1\hspace{0.17em}\cos \left(t\right)+C_2\hspace{0.17em}\sin \left(t\right)$$

zSol(t) = $$C_2\hspace{0.17em}\cos \left(t\right)-C_1\hspace{0.17em}\sin \left(t\right)$$

Solve the differential equation $$\frac{\partial }{\partial t}y(t)=e^{-y(t)}+y(t)$$. dsolve returns an explicit solution in terms of a Lambert W function that has a constant value.

syms y(t)
eqn = diff(y) == y+exp(-y)

eqn(t) = 
$$\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)={\mathrm{e}}^{-y\left(t\right)}+y\left(t\right)$$

sol = dsolve(eqn)

sol = $${\mathrm{W}\text{<a href="matlab:doc symbolic/lambertw">lambertw</a>}}_0\left(-1\right)$$

To return implicit solutions of the differential equation, set the 'Implicit' option to true. An implicit solution has the form $$F(y(t))=g(t)$$.

sol = dsolve(eqn,'Implicit',true)

sol = 
$$\left(\begin{array}{c}\left({\int \frac{{\mathrm{e}}^y}{y\hspace{0.17em}{\mathrm{e}}^y+1}\mathrm{d}y|}_{y=y\left(t\right)}\right)=C_1+t\\ {\mathrm{e}}^{-y\left(t\right)}\hspace{0.17em}\left({\mathrm{e}}^{y\left(t\right)}\hspace{0.17em}y\left(t\right)+1\right)=0\end{array}\right)$$

If dsolve cannot find an explicit solution of a differential equation analytically, then it returns an empty symbolic array. You can solve the differential equation by using MATLAB® numerical solver, such as ode45. For more information, see Solve a Second-Order Differential Equation Numerically.

syms y(x)
eqn = diff(y) == (x-exp(-x))/(y(x)+exp(y(x)));
S = dsolve(eqn)

Warning: Unable to find symbolic solution.

 
S =
 
[ empty sym ]
 

Alternatively, you can try finding an implicit solution of the differential equation by specifying the 'Implicit' option to true. An implicit solution has the form $$F(y(x))=g(x)$$.

S = dsolve(eqn,'Implicit',true)

S = 
$${\mathrm{e}}^{y\left(x\right)}+\frac{{y\left(x\right)}^2}{2}=C_1+{\mathrm{e}}^{-x}+\frac{x^2}{2}$$

Solve the differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\frac{\mathit{a}}{\sqrt{\mathit{y}}}+\mathit{y}$ with condition $$y(a)=1$$. By default, dsolve applies simplifications that are not generally correct, but produce simpler solutions. For more details, see Algorithms.

syms a y(t)
eqn = diff(y) == a/sqrt(y) + y;
cond = y(a) == 1;
ySimplified = dsolve(eqn, cond)

ySimplified = 
$${\left({\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+1\right)}-a\right)}^{2/3}$$

To return the solutions that include all possible values of the parameter $$a$$, turn off simplifications by setting 'IgnoreAnalyticConstraints' to false.

yNotSimplified = dsolve(eqn,cond,'IgnoreAnalyticConstraints',false)

yNotSimplified = 
$$\begin{array}{l}\{\begin{array}{cl}\{\begin{array}{cl}\left\{{\sigma }_1\right\}& \text{ if }-\frac{\pi }{2}<{\sigma }_2\\ \left\{{\sigma }_1,-{\left(-a+{\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+{\left(-\frac{1}{2}+{\sigma }_3\right)}^{3/2}\right)+2\hspace{0.17em}\pi \hspace{0.17em}{C}_2\hspace{0.17em}\mathrm{i}}\right)}^{2/3}\hspace{0.17em}\left(\frac{1}{2}+{\sigma }_3\right)\right\}& \text{ if }{\sigma }_2\le -\frac{\pi }{2}\end{array}& \text{ if }{C}_2\in \mathbb{Z}\\ \varnothing & \text{ if }{C}_2\notin \mathbb{Z}\end{array}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1={\left(-a+{\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+1\right)+2\hspace{0.17em}\pi \hspace{0.17em}{C}_2\hspace{0.17em}\mathrm{i}}\right)}^{2/3}\\ \\ \mathrm{ }{\sigma }_2=\text{angle}\left({\mathrm{e}}^{\frac{3\hspace{0.17em}{C}_1}{2}+\frac{3\hspace{0.17em}t}{2}}-a\right)\\ \\ \mathrm{ }{\sigma }_3=\frac{\sqrt{3}\hspace{0.17em}\mathrm{i}}{2}\end{array}$$

Solve the second-order differential equation $$(x^2-1{)}^2\frac{{\partial }^2}{\partial x^2}y(x)+(x+1)\frac{\partial }{\partial x}y(x)-y(x)=0$$. dsolve returns a solution that contains a term with unevaluated integral.

syms y(x)
eqn = (x^2-1)^2*diff(y,2) + (x+1)*diff(y) - y == 0;
S = dsolve(eqn)

S = 
$$C_2\hspace{0.17em}\left(x+1\right)+C_1\hspace{0.17em}\left(x+1\right)\hspace{0.17em}\int \frac{{\mathrm{e}}^{\frac{1}{2\hspace{0.17em}\left(x-1\right)}}\hspace{0.17em}{\left(1-x\right)}^{1/4}}{{\left(x+1\right)}^{9/4}}\mathrm{d}x$$

To return series solutions of the differential equation around $$x=-1$$, set the 'ExpansionPoint' to -1. dsolve returns two linearly independent solutions in terms of a Puiseux series expansion.

S = dsolve(eqn,'ExpansionPoint',-1)

S = 
$$\left(\begin{array}{c}x+1\\ \frac{1}{{\left(x+1\right)}^{1/4}}-\frac{5\hspace{0.17em}{\left(x+1\right)}^{3/4}}{4}+\frac{5\hspace{0.17em}{\left(x+1\right)}^{7/4}}{48}+\frac{5\hspace{0.17em}{\left(x+1\right)}^{11/4}}{336}+\frac{115\hspace{0.17em}{\left(x+1\right)}^{15/4}}{33792}+\frac{169\hspace{0.17em}{\left(x+1\right)}^{19/4}}{184320}\end{array}\right)$$

Find other series solutions around the expansion point $$\infty$$ by setting 'ExpansionPoint' to Inf.

S = dsolve(eqn,'ExpansionPoint',Inf)

S = 
$$\left(\begin{array}{c}x-\frac{1}{6\hspace{0.17em}{x}^2}-\frac{1}{8\hspace{0.17em}{x}^4}\\ \frac{1}{6\hspace{0.17em}{x}^2}+\frac{1}{8\hspace{0.17em}{x}^4}+\frac{1}{90\hspace{0.17em}{x}^5}+1\end{array}\right)$$

The default truncation order of the series expansion is 6. To obtain more terms in the Puiseux series solutions, set 'Order' to 8.

S = dsolve(eqn,'ExpansionPoint',Inf,'Order',8)

S = 
$$\left(\begin{array}{c}x-\frac{1}{6\hspace{0.17em}{x}^2}-\frac{1}{8\hspace{0.17em}{x}^4}-\frac{1}{90\hspace{0.17em}{x}^5}-\frac{37}{336\hspace{0.17em}{x}^6}\\ \frac{1}{6\hspace{0.17em}{x}^2}+\frac{1}{8\hspace{0.17em}{x}^4}+\frac{1}{90\hspace{0.17em}{x}^5}+\frac{37}{336\hspace{0.17em}{x}^6}+\frac{37}{1680\hspace{0.17em}{x}^7}+1\end{array}\right)$$

Solve the differential equation $\frac{\mathit{dy}}{\mathit{dx}}=\frac{1}{{\mathit{x}}^2}{\mathit{e}}^{-\frac{1}{\mathit{x}}}$ without specifying the initial condition.

syms y(x)
eqn = diff(y) == exp(-1/x)/x^2;
ySol(x) = dsolve(eqn)

ySol(x) = 
$$C_1+{\mathrm{e}}^{-\frac{1}{x}}$$

To eliminate constants from the solution, specify the initial condition $\mathit{y}\left(0\right)=1.$

cond = y(0) == 1;
S = dsolve(eqn,cond)

S = 
$${\mathrm{e}}^{-\frac{1}{x}}+1$$

The function ${\mathit{e}}^{-\frac{1}{\mathit{x}}}$ in the solution ySol(x) has different one-sided limits at $$x=0$$. The function has a right-side limit, $\underset{\mathit{x}\to 0^{+}}{\lim }{\mathit{e}}^{-\frac{1}{\mathit{x}}}=0$, but it has undefined left-side limit, $\underset{\mathit{x}\to 0^{-}}{\lim }{\mathit{e}}^{-\frac{1}{\mathit{x}}}=\infty$.

When you specify the condition y(x0) for a function with different one-sided limits at x0, dsolve treats the condition as a limit from the right, $\lim \mathit{x}\to {\mathit{x}}_0^{+}$.