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Consider the stochastic differential equation of diffusion type driven by Brownian motion dX(t, ω) = μX(t, ω)dt + σX(t, ω)dB (t, ω) where B(t, ω) = limn→∞ B n (t, ω) is a Brownian motion, n is a positive integer, t is time variable, ω is state variable, μ and σ are constants. The solution X(t, ω) is represented by images. Solution contains a term of Brownian motion. Therefore, the image of a solution needs the image of Brownian motion. We have obtained the images of Brownian motion and solution X(t, ω) for different combinations of parameters (μ, σ, n and p. Note that p controls the degree of randomness in Brownian motion. Degree of randomness in Brownian motion is maximum for p = 0.5). The key observations from image analysis are 1. less randomness is visualized for p values away from 0.5, 2. colors in images for n = 10, 000 is more than the color in images for n = 10, 000, 00, and 3. randomness in solution depends on μ and σ also. More randomness is visualized as μ − 1 2 σ 2 is away from 0. The observations are consistent with mathematical analysis of the solution X(t, ω).

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Series solution of rectangular vibrating membrane (SSRVM) of the motion has been represented by Neural Network (NN) model. Experiments have been performed to find a combination of NN structure and size of sample data such that error and time cost of SSRVM are the minimum. Error and time cost of NN models have been studied w.r.t structure, size of sample, random choice of initial weights and selection of data. Peak signal to noise ratio (PSNR) between series solution and solution by NN model is approximately 50 dB. This ensures that solution obtained by NN model is acceptable at human visual system. The average training time of constructed NN models is better than state-of-the-art related to solution of differential equation by NN. NN models are 26 times faster than SSRVM for solution prediction.