# How to solve this system of non-linear differential equations

I'm a bit stumped at how to approach this set of non-linear differential equations. Could someone point me in the right direction in solving them?

System of equations:

$$\dot{p}_{11}(t)=4p_{12}^2(t)+4p_{12}(t)-2$$ $$\dot{p}_{12}(t)=-p_{11}(t)-p_{12}(t)+2p_{22}(t)+4p_{12}(t)p_{22}(t)-3$$ $$\dot{p}_{22}(t)=-2p_{12}(t)-2p_{22}(t)+4p_{22}^2(t)-5$$

Final conditions:

$$p_{11}(5)=1$$ $$p_{12}(5)=0.5$$ $$p_{22}(5)=2$$

Another way to express this system of equations is:

$$f'(t)=4g^2(t)+4g(t)-2$$ $$g'(t)=-f(t)-g(t)+2h(t)+4g(t)h(t)-3$$ $$h'(t)=-2g(t)-2h(t)+4h^2(t)-5$$

Where

$$f(5)=1$$ $$g(5)=0.5$$ $$h(5)=2$$

Thank you all in advance for your help. I deeply appreciate it.

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You can have a numerical solution to your system. Here are the values of the functions at $t=0$
$$[t= 0,f \left( t \right) = 1.736079,g \left( t \right) = 0.3659732,h\left( t \right) = 1.472899]$$
Notice that, for $t>5$, it seems there is a singularity. The three functions, $f(t),g(t),h(t)$ have the following plots respectively