# Using numerical methods to plot solution to first-order nonlinear differential equation

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Aleem Andrew on 6 Nov 2021
Edited: David Goodmanson on 17 Nov 2021
I have a question about plotting x(t), the solution to the following differential equation knowing that dx/dt equals the expression below. The value of x is 0 at t = 0.
syms x
dxdt = -(1.0*(6.84e+45*x^2 + 5.24e+32*x - 2.49e+42))/(2.47e+39*x + 7.12e+37)
I want to plot the solution of this first-order nonlinear differential equation. The analytical solution involves complex numbers so that's not relevant but Matlab can solve the equation using numerical methods and plot it. Can anyone please suggest how to do this? Thank you.
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Sahil Jain on 16 Nov 2021
The solution given by @Chris seems correct. For more information, you can refer to the documentation page for "ode45".

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### Answers (1)

David Goodmanson on 17 Nov 2021
Edited: David Goodmanson on 17 Nov 2021
Hello Aleem,
You are saying that the analytical solution is not relevant because it involves complex numbers, but that's not the case. We have
dx/dt = f(x) --> t = Integral 1/f(x) dx + const
In code,
syms x
t = int(1/(-(1.0*(6.84e+45*x^2 + 5.24e+32*x - 2.49e+42))/(2.47e+39*x + 7.12e+37)))
t = vpa(t,6)
t = 9.22306e-8*log(x + 0.0190797) - 4.53342e-7*log(x - 0.0190797)
For x near 0, t is complex because the argument of the second log term is negative. But
log(-a) = log(+a) + i*pi,
so a solution that is just as good can be obtained by subtracting off the constant i*pi, resulting in a reversed sign for of the argument of the second term. Then the function is real for 0 < x < 0.0190797. So
fun2 = @(x) 9.22306e-8*log(x + 0.0190797) - 4.53342e-7*log(0.0190797 - x);
y = linspace(0,.0190796,1000); % pick upper limit for y that is just less than 0.0190797
t = fun2(y)-fun2(0); % subtract a constant so that t meets the boundary condition
% analytic solution
ya = y;
ta = t;
% compare with Chris's solution
fun = @(t,x) -(1.0.*(6.84e+45.*x.^2 + 5.24e+32.*x - 2.49e+42))./(2.47e+39.*x + 7.12e+37);
tspan = [0,5e-6];
x0 = 0;
[t,y] = ode45(fun, tspan,x0);
figure(1)
plot(t,y,ta,ya+.0002) % add a small offset, otherwise they overlay
grid on
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