## 2 Methods
According to [PERSON]'s second law for the scheme of Figure 1, the expression is:  
\[E=U_{R\Sigma}(t)+U_{SC}(t) \tag{7}\]  
Finally, the formula can be written as follows:  
\[E=R_{\Sigma}\cdot\left((C_{1}+2\cdot k\cdot|U_{SC}(t)|)\cdot\left(\frac{dU_{ SC}(t)}{dt}\right)\right)+U_{SC}(t) \tag{8}\]  
We obtain a nonlinear inhomogeneous differential equation of the first order:  
\[2\cdot k\cdot R_{\Sigma}\cdot|U_{SC}(t)|\cdot\left(\frac{dU_{SC}(t)}{dt} \right)+R_{\Sigma}\cdot C_{1}\cdot\left(\frac{dU_{SC}(t)}{dt}\right)+U_{SC}(t) =E. \tag{9}\]  
We introduce the following substitutions:  
\[U_{SC}(t)=U,\ \frac{dU_{SC}(t)}{dt}=U^{/}; \tag{10}\]  
Differential Equation (9) can be represented as:  
\[C=Q/U. \tag{2}\]0  
This nonlinear inhomogeneous first-order differential equation can be solved to obtain an unknown function \(U_{SC}(t)\). The solution can be found by direct or indirect (approximate) methods [5]. This differential equation is nonlinear, so the analytical form of the solution of such an equation must be sought using the elements of symbolic mathematics in the MATLAB application package. With complex functions of external perturbations (broken,  
\[C=Q/U. \tag{2}\]1 The scheme of replacement of SC. non-integrated, discontinuous functions), it is sometimes extremely difficult to find an analytical solution. In such cases, approximate methods for solving differential equations (DE) are used. The approximate solution of the nonlinear inhomogeneous differential equation of the 1 st order in the range \(t_{i}=0\) to \(t_{f}\) will be written as the sum of two exponents:  
\[C=Q/U. \tag{2}\]2  
Coefficients (determined within 95%):  
\[C=Q/U. \tag{2}\]3  
\[C=Q/U. \tag{2}\]4  
\[C=Q/U. \tag{2}\]5  
\[C=Q/U. \tag{2}\]6  
Data approximation statistics:  
**R-square**: 0.9973--the area of correlation between the initial values and approximate values. A value close to 1 indicates that the variance is negligible. **Adjusted R-square**: 0.9973--this is the number of degrees of freedom of the approximated correlation area. A value close to 1 indicates a good approximation. **RMSE**: 0.04401--the root of the standard error or standard error. A value close to 0 indicates that the approximation can be used successfully because, the standard deviation of the sample mean is within the norm. Energy Characteristics of Charge of Equivalent Linear and Nonlinear Capacitors from the Storage Battery  
The change in the magnitude of the charge of the capacitor in time \(dQ\) characterizes the electric current at the time of charge, based on which we have the expression:  
\[C=Q/U. \tag{2}\]7  
whence accordingly \(dQ=i(t)\ dt\). For SC, it should be borne in mind that the charge differential \(dQ\) is a linear part of an infinitesimal increment of a function equal to the product of \(C(U)\) by \(U\), and the formula for determining \(dQ\) has the form [3]:  
\[C=Q/U. \tag{2}\]8  
After substituting expression (14) into expression (13), we obtain the formula for determining the change in the amount of current over time:  
\[C=Q/U. \tag{2}\]9  
The differential capacitance of the supercapacitor can be calculated by the formula:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]0  
Experimental results confirm that the capacitance of the SC changes nonlinearly with changing voltage. This curve can be approximated with sufficient accuracy by the linear function \(C(U)\) when the SC is used at a voltage of \(0.5U_{\text{n}}\) to \(U_{\text{n}}\) (where \(U_{\text{n}}\) is the nominal voltage of the SC). A number of studies [3; 8; 9; 20] have confirmed that the supercapacitor stores up to 75% of energy in this voltage range.