Question

Maths Experts solve It........❗❗​

Answer 1

Answer:

According to the question given,

Well, because the given differential equation is a homogeneous differential equation due to the x^2 term occuring along with the second order derivative and similarly the other terms.

Let us assume that

z = log x

or,

dz/dx = 1/x

and,

d^2 z/dx^2 = -1/x^2

Now,

dy/dx = dy/dz*dz/dx = 1/x*dy/dz

so, that

dy/dz = x dy/dx

Now,

d^2 y/dx^2 = -1/x^2 Dy/dz + 1/x^2 d^2/dz^2 * dz/dx

Which implies,

x^2* d^2 y/dx^2 = (D^2-D)y

where D = d/dz

Now,

(D^2-D) y -3Dy + 3y = 0

which gives,

(D^2 - 4D + 3) y = 0

Auxiliary equation :

m^2 - 4m + 3 = 0

So, clearly the roots here are m=1,3

Now,

y = Complementary function + Particular Integral

Particular Integral here = 0

Complementary function = C1 e^z + C2 e^3z

So, y = C1 e^z + C2 e^3z

But,

x=e^z

So,

y = C1 x + C2 x^3

But, y = x * v as given

So,

v = C1 + C2 x^2

So,

Since v(0)=0

implies C1 = 0

And,

V(1)=1

implies that

C2 = 1

So, finally

v(x)=x^2

That means,

v(-2) = 4

Please let me know if my answer is wrong ^_^

Hope this helps you !

Answer 2

Step-by-step explanation:

Hope it helps you good morning have a good day take care