This help page gives a few examples of using the command ODESteps to solve first order ordinary differential equations.

See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right)\:$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=0$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)=0$

$\mathrm{ODESteps}\left(\mathrm{ode1}\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)=0\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆt}z\left(t\right)\\ \text{•}& & \text{Solve for the highest derivative}\\ & & \frac{\ⅆ}{\ⅆt}z\left(t\right)=-\frac{t^2\left(z\left(t\right)+1\right)}{{z\left(t\right)}^2\left(t-1\right)}\\ \text{•}& & \text{Separate variables}\\ & & \frac{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right){z\left(t\right)}^2}{z\left(t\right)+1}=-\frac{t^2}{t-1}\\ \text{•}& & \text{Integrate both sides with respect to}t\\ & & \int \frac{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right){z\left(t\right)}^2}{z\left(t\right)+1}\ⅆt=\int -\frac{t^2}{t-1}\ⅆt+\mathrm{c\_\_1}\\ \text{•}& & \text{Evaluate integral}\\ & & \frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+\mathrm{c\_\_1}\end{array}$

$\mathrm{ode2}≔\frac{\mathrm{diff}\left(y\left(x\right)\,x\right)}{f\left(x\right)}+y\left(x\right)+g\left(x\right)=0$

$\mathrm{ode2}≔\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{f\left(x\right)}+y\left(x\right)+g\left(x\right)=0$

$\mathrm{ODESteps}\left(\mathrm{ode2}\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & \frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{f\left(x\right)}+y\left(x\right)+g\left(x\right)=0\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)\\ \text{•}& & \text{Solve for the highest derivative}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=-\left(y\left(x\right)+g\left(x\right)\right)f\left(x\right)\\ \text{•}& & \text{Collect w.r.t.}y\left(x\right)\text{and simplify}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=-f\left(x\right)y\left(x\right)-f\left(x\right)g\left(x\right)\\ \text{•}& & \text{Group terms with}y\left(x\right)\text{on the lhs of the ODE and the rest on the rhs of the ODE}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)=-f\left(x\right)g\left(x\right)\\ \text{•}& & \text{The ODE is linear; multiply by an integrating factor}\mathrm{\mu }\left(x\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)\right)=-\mathrm{\mu }\left(x\right)f\left(x\right)g\left(x\right)\\ \text{•}& & \text{Assume the lhs of the ODE is the total derivative}\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)\right)=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\mathrm{\mu }\left(x\right)+y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\right)\\ \text{•}& & \text{Isolate}\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\\ & & \frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)=\mathrm{\mu }\left(x\right)f\left(x\right)\\ \text{•}& & \text{Solve to find the integrating factor}\\ & & \mathrm{\mu }\left(x\right)={\ⅇ}^{\int f\left(x\right)\ⅆx}\\ \text{•}& & \text{Integrate both sides with respect to}x\\ & & \int \left(\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\right)\ⅆx=\int -\mathrm{\mu }\left(x\right)f\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Evaluate the integral on the lhs}\\ & & y\left(x\right)\mathrm{\mu }\left(x\right)=\int -\mathrm{\mu }\left(x\right)f\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Solve for}y\left(x\right)\\ & & y\left(x\right)=\frac{\int -\mathrm{\mu }\left(x\right)f\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}}{\mathrm{\mu }\left(x\right)}\\ \text{•}& & \text{Substitute}\mathrm{\mu }\left(x\right)={\ⅇ}^{\int f\left(x\right)\ⅆx}\\ & & y\left(x\right)=\frac{\int -{\ⅇ}^{\int f\left(x\right)\ⅆx}f\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}}{{\ⅇ}^{\int f\left(x\right)\ⅆx}}\\ \text{•}& & \text{Simplify}\\ & & y\left(x\right)={\ⅇ}^{-\int f\left(x\right)\ⅆx}\left(-\int {\ⅇ}^{\int f\left(x\right)\ⅆx}f\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}\right)\end{array}$

$\mathrm{ode3}≔\frac{\mathrm{diff}\left(y\left(x\right)\,x\right)}{y\left(x\right)}+1+\frac{g\left(x\right)}{y\left(x\right)}=0$

$\mathrm{ode3}≔\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{y\left(x\right)}+1+\frac{g\left(x\right)}{y\left(x\right)}=0$

$\mathrm{ODESteps}\left(\mathrm{ode3}\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & \frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{y\left(x\right)}+1+\frac{g\left(x\right)}{y\left(x\right)}=0\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)\\ \text{•}& & \text{Solve for the highest derivative}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=\left(-1-\frac{g\left(x\right)}{y\left(x\right)}\right)y\left(x\right)\\ \text{•}& & \text{Collect w.r.t.}y\left(x\right)\text{and simplify}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=-y\left(x\right)-g\left(x\right)\\ \text{•}& & \text{Group terms with}y\left(x\right)\text{on the lhs of the ODE and the rest on the rhs of the ODE}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)+y\left(x\right)=-g\left(x\right)\\ \text{•}& & \text{The ODE is linear; multiply by an integrating factor}\mathrm{\mu }\left(x\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+y\left(x\right)\right)=-\mathrm{\mu }\left(x\right)g\left(x\right)\\ \text{•}& & \text{Assume the lhs of the ODE is the total derivative}\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+y\left(x\right)\right)=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\mathrm{\mu }\left(x\right)+y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\right)\\ \text{•}& & \text{Isolate}\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\\ & & \frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)=\mathrm{\mu }\left(x\right)\\ \text{•}& & \text{Solve to find the integrating factor}\\ & & \mathrm{\mu }\left(x\right)={\ⅇ}^x\\ \text{•}& & \text{Integrate both sides with respect to}x\\ & & \int \left(\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\right)\ⅆx=\int -\mathrm{\mu }\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Evaluate the integral on the lhs}\\ & & y\left(x\right)\mathrm{\mu }\left(x\right)=\int -\mathrm{\mu }\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Solve for}y\left(x\right)\\ & & y\left(x\right)=\frac{\int -\mathrm{\mu }\left(x\right)g\left(x\right)\ⅆx+\mathrm{c\_\_1}}{\mathrm{\mu }\left(x\right)}\\ \text{•}& & \text{Substitute}\mathrm{\mu }\left(x\right)={\ⅇ}^x\\ & & y\left(x\right)=\frac{\int -{\ⅇ}^{x}g\left(x\right)\ⅆx+\mathrm{c\_\_1}}{{\ⅇ}^x}\\ \text{•}& & \text{Simplify}\\ & & y\left(x\right)={\ⅇ}^{-x}\left(-\int {\ⅇ}^{x}g\left(x\right)\ⅆx+\mathrm{c\_\_1}\right)\end{array}$

$\mathrm{ode4}≔2xy\left(x\right)-9x^2+\left(2y\left(x\right)+x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode4}≔2xy\left(x\right)-9x^2+\left(2y\left(x\right)+x^2+1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0$

$\mathrm{ODESteps}\left(\mathrm{ode4}\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & 2xy\left(x\right)-9x^2+\left(2y\left(x\right)+x^2+1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)\\ \text{▫}& & \text{Check if ODE is exact}\\ & \text{◦}& \text{ODE is exact if the lhs is the total derivative of a}{C}^2\text{function}\\ & & \frac{\ⅆ}{\ⅆx}G\left(x\,y\left(x\right)\right)=0\\ & \text{◦}& \text{Compute derivative of lhs}\\ & & \frac{\partial }{\partial x}G\left(x\,y\right)+\left(\frac{\partial }{\partial y}G\left(x\,y\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0\\ & \text{◦}& \text{Evaluate derivatives}\\ & & 2x=2x\\ & \text{◦}& \text{Condition met, ODE is exact}\\ \text{•}& & \text{Exact ODE implies solution will be of this form}\\ & & \left[G\left(x\,y\right)=\mathrm{c\_\_1}\,M\left(x\,y\right)=\frac{\partial }{\partial x}G\left(x\,y\right)\,N\left(x\,y\right)=\frac{\partial }{\partial y}G\left(x\,y\right)\right]\\ \text{•}& & \text{Solve for}G\left(x\,y\right)\text{by integrating}M\left(x\,y\right)\text{with respect to}x\\ & & G\left(x\,y\right)=\int \left(-9x^2+2xy\right)\ⅆx+\mathrm{\_F1}\left(y\right)\\ \text{•}& & \text{Evaluate integral}\\ & & G\left(x\,y\right)=-3x^3+x^{2}y+\mathrm{\_F1}\left(y\right)\\ \text{•}& & \text{Take derivative of}G\left(x\,y\right)\text{with respect to}y\\ & & N\left(x\,y\right)=\frac{\partial }{\partial y}G\left(x\,y\right)\\ \text{•}& & \text{Compute derivative}\\ & & x^2+2y+1=x^2+\frac{\ⅆ}{\ⅆy}\mathrm{\_F1}\left(y\right)\\ \text{•}& & \text{Isolate for}\frac{\ⅆ}{\ⅆy}\mathrm{\_F1}\left(y\right)\\ & & \frac{\ⅆ}{\ⅆy}\mathrm{\_F1}\left(y\right)=2y+1\\ \text{•}& & \text{Solve for}\mathrm{\_F1}\left(y\right)\\ & & \mathrm{\_F1}\left(y\right)=y^2+y\\ \text{•}& & \text{Substitute}\mathrm{\_F1}\left(y\right)\text{into equation for}G\left(x\,y\right)\\ & & G\left(x\,y\right)=-3x^3+x^{2}y+y^2+y\\ \text{•}& & \text{Substitute}G\left(x\,y\right)\text{into the solution of the ODE}\\ & & -3x^3+x^{2}y+y^2+y=\mathrm{c\_\_1}\\ \text{•}& & \text{Solve for}y\left(x\right)\\ & & \left\{y\left(x\right)=-\frac{x^2}{2}-\frac{1}{2}-\frac{\sqrt{x^4+12x^3+2x^2+4\mathrm{c\_\_1}+1}}{2}\,y\left(x\right)=-\frac{x^2}{2}-\frac{1}{2}+\frac{\sqrt{x^4+12x^3+2x^2+4\mathrm{c\_\_1}+1}}{2}\right\}\end{array}$

$\mathrm{ode5}≔\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)-x\exp \left(x\right)=0$

$\mathrm{ode5}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)-x{\ⅇ}^x=0$

$\mathrm{ODESteps}\left(\mathrm{ode5}\right)$

$\begin{array}{lll}& & \text{Let's solve}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)-x{\ⅇ}^x=0\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)\\ \text{•}& & \text{Solve for the highest derivative}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=y\left(x\right)+x{\ⅇ}^x\\ \text{•}& & \text{Group terms with}y\left(x\right)\text{on the lhs of the ODE and the rest on the rhs of the ODE}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)=x{\ⅇ}^x\\ \text{•}& & \text{The ODE is linear; multiply by an integrating factor}\mathrm{\mu }\left(x\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)\right)=\mathrm{\mu }\left(x\right)x{\ⅇ}^x\\ \text{•}& & \text{Assume the lhs of the ODE is the total derivative}\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\\ & & \mathrm{\mu }\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)\right)=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\mathrm{\mu }\left(x\right)+y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\right)\\ \text{•}& & \text{Isolate}\frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)\\ & & \frac{\ⅆ}{\ⅆx}\mathrm{\mu }\left(x\right)=-\mathrm{\mu }\left(x\right)\\ \text{•}& & \text{Solve to find the integrating factor}\\ & & \mathrm{\mu }\left(x\right)={\ⅇ}^{-x}\\ \text{•}& & \text{Integrate both sides with respect to}x\\ & & \int \left(\frac{\ⅆ}{\ⅆx}\left(y\left(x\right)\mathrm{\mu }\left(x\right)\right)\right)\ⅆx=\int \mathrm{\mu }\left(x\right)x{\ⅇ}^x\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Evaluate the integral on the lhs}\\ & & y\left(x\right)\mathrm{\mu }\left(x\right)=\int \mathrm{\mu }\left(x\right)x{\ⅇ}^x\ⅆx+\mathrm{c\_\_1}\\ \text{•}& & \text{Solve for}y\left(x\right)\\ & & y\left(x\right)=\frac{\int \mathrm{\mu }\left(x\right)x{\ⅇ}^x\ⅆx+\mathrm{c\_\_1}}{\mathrm{\mu }\left(x\right)}\\ \text{•}& & \text{Substitute}\mathrm{\mu }\left(x\right)={\ⅇ}^{-x}\\ & & y\left(x\right)=\frac{\int {\ⅇ}^{-x}x{\ⅇ}^x\ⅆx+\mathrm{c\_\_1}}{{\ⅇ}^{-x}}\\ \text{•}& & \text{Evaluate the integrals on the rhs}\\ & & y\left(x\right)=\frac{\frac{x^2}{2}+\mathrm{c\_\_1}}{{\ⅇ}^{-x}}\\ \text{•}& & \text{Simplify}\\ & & y\left(x\right)={\ⅇ}^x\left(\frac{x^2}{2}+\mathrm{c\_\_1}\right)\end{array}$