Find integrable combination for solving system of differential equations

Question

$$ \left\{ \begin{array}{c} \dot{x}_1 = 2t(x_1+t^2) \\ \dot{x}_2 = t(x_1^2-t^4-2t^2-1)+x_2^2 \end{array} \right. $$

$$ \left\{ \begin{array}{c} x_1(1)=-2 \\ x_2(1) = -1 \end{array} \right. $$

the way this system should be solved is to find an integrable combination there is the description (page 349 of pdf document).

I've already tried multiplying the first equation by $x_1$, the second one by $x_2$ then adding first to second so I got nothing solvable.

Also I tried multiplying the first by $x_2$, the second by $x_1$, also adding first to second so I got also nothing.

Could you please provide some integrating combinations.

Answer 1

1. We solve first equation (it's independent of the second, Bernoulli equation) and get:

$$x_1(t)=c_1e^{t^2}-t^2-1$$

2. Then we substitute $x_1(1)=-2$ and get $c_1= 0$ hence: $$x_1(t)=-t^2-1$$
3. Substitute $x_1(t)$ to the second equation and get $$\dot{x}_2(t)=x_2^2$$
4. We solve this equation so we have: $$x_2(t)=-\frac{1}{t+c_2}$$
5. Finally we substitute the $x_2(1)=-1$ and get the answer: $$x_2(t)=\frac{-1}{t}$$