Is it possible that a single neural network structure can solve multiple problems?

I am an enthusiast of neural network and regularly I try to model simple models in NN. While thinking about neural network's application in artificial intelligence I had this doubt arise in my mind whether a single neural network can solve multiple tasks or problems. A neural network trained to classify digits from images can also be used "simultaneously" for other task such as writing an original article. I know we can retrain same neural network depending on the task but I am not looking for that. In order to check my doubt I created a neural network which should in theory will solve two continuous equations (will add more equations in future) and this is my code

library(caret)
set.seed(2016)
#first equation
equ1<-function(x){
v=x^3/(2*tan(x))
return(v)
}
#second equation
equ2<-function(x){
v=sin(x^2)/2*x
return(v)
}
X=(1:50)
Y1=equ1(X)
Y2=equ2(X)
I=cbind(X,Y1,Y2)
I=data.frame(I)

ctrl <- trainControl(method = "cv",number=10, verboseIter = TRUE, savePred=T)
model <- train(Y1+Y2~X, data=I, method ="nnet", trControl = ctrl,verbose = TRUE)

print(model)


and the out put is this

  size  decay  RMSE      Rsquared    RMSE SD   Rsquared SD
1     0e+00  187737.1         NaN  373322.1         NA
1     1e-04  187737.1         NaN  373322.1         NA
1     1e-01  187737.1  0.09728231  373322.1         NA
3     0e+00  187737.1         NaN  373322.1         NA
3     1e-04  187737.1         NaN  373322.1         NA
3     1e-01  187737.1  0.15565127  373322.1  0.1361369
5     0e+00  187737.1         NaN  373322.1         NA
5     1e-04  187737.1         NaN  373322.1         NA
5     1e-01  187737.1  0.30146221  373322.1  0.2960712


From the result we can see that its RMSE value is way greater and non of the models can be used for solving the two equations?

1Q.Does a single neural network can solve multiple tasks?
2Q.If yes,Did I made any error in my code which is giving me negative results?
3Q.Can anyone share any neural network example which can be successfully used for different tasks(without retraining them)?


Yes, a network can perform multiple tasks. Here's a simple example. Say network $N_1$ computes function $f_1$ and network $N_2$ computes function $f_2$. Both contain $n_{in}$ input units. Construct a new network $N_3$ with $n_{in}$ input units. The hidden and output layers of $N_3$ are side-by-side, concatenated copies of those of $N_1$ and $N_2$. So, $N_3$ has two sets of outputs. The first set (matching $N_1$) will return the output of $f_1$. The second set (matching $N_2$) will return the output of $f_2$.
That describes how to construct a new, multi-function network out of individual, trained, single-function networks. However, it should also be possible to train a multi-function network $N_4$ from the ground up. Such a network would have multiple sets of outputs (one for each function), and the loss function would be written to take this into account. Network $N_3$ (above) consists of parallel, independent subnetworks; the weight matrices have a block structure such that the hidden units within each subnetwork don't interact. But, the functions computed by $N_4$ might end up 'sharing' hidden units.