INTEGRAL TRANSFORMS WITH APPLICATIONS

FOURTH SEMESTER – Mathematics
Regular & Supplementary – March/April 2026
Time: 3 Hours | Max. Marks: 75

SECTION – A

Answer any FIVE questions. Each answer carries 5 marks.
5 × 5 = 25 Marks

1. Solve [formula is partially garbled in the PDF extraction].
2. Solve $$(D^2-2D+2)y=0$$ given the initial conditions shown in the paper.
3. Solve $$ty”+2y’+ty=0$$ with the given initial conditions.
4. Solve the simultaneous differential equations $$-2u+\cdots$$ [the remaining expression is not readable in the extracted PDF].
5. Convert $$y”(t)-3y'(t)+2y(t)=4\sin t$$ with $$y(0)=1,\qquad y'(0)=-2$$ into an integral equation.
6. State and prove the Modulation Theorem for Fourier Transforms.
7. Find the Fourier sine transform of $$f(x)=\cdots$$ [function is not legible in the extracted text].
8. Solve the integral equation $$\int \cdots f(x)\cos(\lambda x)\,dx=\cdots$$ [formula is partially garbled].

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SECTION – B

Answer FIVE of the following questions. Each question carries 10 marks.
5 × 10 = 50 Marks

UNIT – I

9.

(a) Solve$$(D^2-3D+2)y=1-e^{\cdots}$$

given$$y(0)=1,\qquad Dy(0)=0.$$

OR

(b) Solve$$ty”+y’+4ty=0$$

given$$y(0)=3,\qquad y'(0)=0.$$

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UNIT – II

10.

(a) Solve the differential equation shown in the paper:$$D^2y+Dy=t,\qquad \cdots$$

OR

(b) Solve$$\cdots=xt$$

given$$y=0$$

[the PDF extraction has lost part of the equation].

11.

Solve the integral equation$$\cdots$$

[formula is not readable in the extracted text].

12.

(a) Show that$$\cdots$$

[expression is incomplete in the PDF text extraction].

(b) If$$x(0)=3,\qquad x'(0)=-2,\qquad y(0)=0,$$

solve the given system/equation.

OR

Using Laplace transform, solve$$y(t)=\cdots+\int_0^t y(t-u)\sin u\,du.$$

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UNIT – III

The extracted PDF shows a question involving an integral expression and another problem with boundary/initial conditions, but several mathematical symbols are missing from the text extraction.

13.

State and prove the Change of Scale Property for Fourier Transforms.

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UNIT – IV

14.

Find the Fourier cosine transform of$$e^{\cdots}$$

[exponent/function is incomplete in extraction].

OR

State and prove Parseval’s Identity for Transforms.

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UNIT – V

The paper’s final question is shown under UNIT–V, but the actual mathematical expression is not captured by the PDF text extraction.