Q.1 Solve the given differential equation subject to the indicated initial condition.
xy(1+xy^2)dy/dx = 1, y(1)=0

Q.2 Find the orthogonal trajectories of the following family of curves.
y^3+3x^2y=c1

Q.3 Determine whether the functions f1(x)=x, f2(x)=x ln x, f3(x) = x^2lnx are linearly independent or linearly dependent on (0,∞).

Q.1 Solve the given differential equation subject to the indicated initial condition.
xy(1+xy^2)dy/dx = 1, y(1)=0

Is there a possible typo? Just asking because this is really not doable by the usual means. I even ran it through Maple for a check and it gave me a crazy solution involving the Lambert W function. What section of DE did this come from? Bernoulli? Separable? Non-Homogeneous? Ricatti? Something else?


Q.2 Find the orthogonal trajectories of the following family of curves.
y^3+3x^2y=c1

If f(x,y)=dy/dx is the DE of one family, then the DE for the orthogonal trajectories of the other family is dy/dx=-1/f(x,y).

Suppose we have y=-x-1+Ce^{x}. Then, the DE is

y'=x+y. Thus, the DE of the orthogonal family is

\frac{dy}{dx}=\frac{-1}{x+y}\Rightarrow \frac{dx}{dy}+x=-y

It may be easier to find dx/dy rather than dy/dx.

From the given function, by implicit differentiation w.r.t x, we get

\frac{dx}{dy}=\frac{-(x^{2}+y^{2})}{2xy}

Now, find the orthogonal by using the info I gave.


Q.3 Determine whether the functions f1(x)=x, f2(x)=x ln x, f3(x) = x^2lnx are linearly independent or linearly dependent on (0,∞).

I can not read what interval you mean?

But, the functions are linearly independent when neither is a constant multiple of the other on an interval.

For instance, f_{1}(x)=\sqrt{x}+5, \;\ f_{2}(x)=\sqrt{x}+5x, \;\ f_{3}(x)=x-1, \;\ f_{4}(x)=x^{2}

These are linearly dependent on the interval (0,{\infty}) since f_{2}(x) can be written as a linear combo of f_{1}, \;\ f_{3}, \;\ f_{4}

f_{2}(x)=1\cdot f_{1}(x)+5f_{3}(x)+0\cdot f_{4}(x)

You can also check the Wronskian. It is linearly independent if the Wronskian is not equal to 0 for every x in the interval.

A Wronskian is the determinant of the matrix made up of the functions and their first and second derivatives.

I get W=xln(x)+2x

Does this equal 0 on the interval?

Q.1 comes from Bernoulli section...
And I can't type Q.3 in right way but it is in the same format that you have given. Looking for further help. Thanks.