**Q.1 a i)** Solve

**Solution** : The auxiliary equation is or . The roots are and . The complimentary function will be

The particular integral will be

Using ,

Using ,

Therefore,

The complete solution is

**Q.1 a ii)** Solve

**Solution** : These are symmetric simultaneous differential equations. We will find 2 sets of values which will give .

Note the pattern in the ratios. In the first ratio, and are absent in the denominator, in the second, and and in the third and .

If we choose the first set of multipliers as and , we get,

Hence,

On integrating,

Note that this represents a **family of planes** normal to the position vector of the point .

If we choose the second set of multipliers as and , we get

Hence,

On integrating,

Note that this represents a **family of spheres** with center at origin.

**Q.1 a iii)** Solve

**Solution** : The auxiliary equation is . Its roots are and . These are complex conjugate of each other. Hence, the complimentary function will be

The particular integral will be

Note that first two functions on RHS are algebraic functions with integral powers of . Hence, we need to use the binomial expansion in the form .

The third function is . If we use , we get , so we need to use the alternative, which is .

Considering all these,

All derivatives of of order will be and all derivatives of of order will be . So,

The complete solution is