Define a simple ODE.

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=2y\left(x\right)+1$

$\mathrm{ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=2y\left(x\right)+1$

Solve the ODE, ode.

$\mathrm{dsolve}\left(\mathrm{ode}\right)$

$y\left(x\right)={\ⅇ}^{\sqrt{2}x}\mathrm{\_C2}+{\ⅇ}^{-\sqrt{2}x}\mathrm{\_C1}-\frac{1}{2}$

Define initial conditions.

$\mathrm{ics}≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=0$

$\mathrm{ics}≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=0$

Solve ode subject to the initial conditions ics.

$\mathrm{dsolve}\left(\left\{\mathrm{ics}\,\mathrm{ode}\right\}\right)$

$y\left(x\right)=\frac{3{\ⅇ}^{\sqrt{2}x}}{4}+\frac{3{\ⅇ}^{-\sqrt{2}x}}{4}-\frac{1}{2}$

Compute the solution using the Laplace transform method.

$\mathrm{sol}≔\mathrm{dsolve}\left(\left\{\mathrm{ics}\,\mathrm{ode}\right\}\,y\left(x\right)\,\mathrm{method}=\mathrm{laplace}\right)$

$\mathrm{sol}≔y\left(x\right)=-\frac{1}{2}+\frac{3\cosh \left(\sqrt{2}x\right)}{2}$

Test whether the ODE solution satisfies the ODE and the initial conditions (see odetest).

$\mathrm{odetest}\left(\mathrm{sol}\,\left[\mathrm{ode}\,\mathrm{ics}\right]\right)$

$\left[0\,0\,0\right]$

Find a series solution for the same problem.

$\mathrm{series\_sol}≔\mathrm{dsolve}\left(\left\{\mathrm{ics}\,\mathrm{ode}\right\}\,y\left(x\right)\,\mathrm{series}\right)$

$\mathrm{series\_sol}≔y\left(x\right)=1+\frac{3}{2}{x}^2+\frac{1}{4}{x}^4+O\left(x^6\right)$

$\mathrm{odetest}\left(\mathrm{series\_sol}\,\left[\mathrm{ode}\,\mathrm{ics}\right]\,\mathrm{series}\right)$

$\left[0\,0\,0\right]$

Define a system of ODEs.

$\mathrm{sys\_ode}≔\mathrm{diff}\left(y\left(t\right)\,t\right)=x\left(t\right),\mathrm{diff}\left(x\left(t\right)\,t\right)=-x\left(t\right)$

$\mathrm{sys\_ode}≔\frac{\ⅆ}{\ⅆt}y\left(t\right)=x\left(t\right),\frac{\ⅆ}{\ⅆt}x\left(t\right)=-x\left(t\right)$

If the unknowns are not specified, all differentiated indeterminate functions in the system are treated as the unknowns of the problem.

$\mathrm{dsolve}\left(\left[\mathrm{sys\_ode}\right]\right)$

$\left\{x\left(t\right)=\mathrm{\_C2}{\ⅇ}^{-t}\,y\left(t\right)=-\mathrm{\_C2}{\ⅇ}^{-t}+\mathrm{\_C1}\right\}$

Define initial conditions.

$\mathrm{ics}≔x\left(0\right)=1,y\left(1\right)=0$

$\mathrm{ics}≔x\left(0\right)=1,y\left(1\right)=0$

Solve the system of ODEs subject to the initial conditions ics.

$\mathrm{dsolve}\left(\left[\mathrm{sys\_ode}\,\mathrm{ics}\right]\right)$

$\left\{x\left(t\right)={\ⅇ}^{-t}\,y\left(t\right)=-{\ⅇ}^{-t}+{\ⅇ}^{\mathrm{-1}}\right\}$