BMTC-132 Solved Assignment January 2026 | DIFFERENTIAL EQUATIONS | IGNOU BSCG

Q: State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:

Answer available — download the full PDF for complete details.

Q: No main question text, refer to sub-questions.

Answer available — download the full PDF for complete details.

Q: No main question text, refer to sub-questions.

Answer available — download the full PDF for complete details.

Q: No main question text, refer to sub-questions.

Answer available — download the full PDF for complete details.

Q1. State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:

a)) y² is an integrating factor of the differential equation: 6xy dx + (4y +9x²)dy = 0. (75 words)
b)) The solution of the differential equation dy/d = y with y(0) = 0 exists, but is not unique. (75 words)
c)) sin x d²y/dx² + dy/dx + y = 0 in ]0, π[ is a linear homogeneous equation. (75 words)
d)) The solution of the differential equation dy/dx = y with y(0) = 0 exists, but is not unique. (75 words)
e)) The Pfaffian equation (2xy² + 2xy + 2xz² + 1)dx + dy + 2z dz = 0 is integrable. (75 words)

Key Points:

Integrating factor transforms a non-exact DE into an exact one.
Exactness condition for M dx + N dy = 0 is ∂M/∂y = ∂N/∂x.
Existence and Uniqueness Theorem guarantees a unique solution if f(x,y) and ∂f/∂y are continuous.
A linear differential equation has dependent variable and its derivatives to the first power only.

Answer:

Q2. No main question text, refer to sub-questions.

a)) Apply the method of variation of parameter to solve the differential equation: y'' +6y'+9y = -1/x e^(-3x), x > 0. (175 words)
b)) Suppose that a thermometer having a reading of 75°F inside a house is placed outside where the air temperature is 15°F. Two minutes later it is found that the thermometer reading is 30°F. Find the temperature reading T(t) of the thermometer at any time t. (175 words)

Key Points:

Variation of parameters uses the Wronskian to find a particular solution for non-homogeneous DEs.
The Wronskian W(e^(-3x), xe^(-3x)) for the given problem is e^(-6x).
Particular solution yp for y''+6y'+9y = -1/x e^(-3x) is xe^(-3x) - xe^(-3x)ln(x).
Newton's Law of Cooling is modeled by the differential equation dT/dt = k(T - Ts).

Answer:

Q3. No main question text, refer to sub-questions.

a)) Find the integral surface of the p.d.e.: (x - y)p + (y - x - z)q = z through the circle z = 1, x² + y² = 1. (175 words)
b)) Solve: (x²y-2xy²)dx – (x³-3x²y)dy = 0. (175 words)

Key Points:

Lagrange's method solves P p + Q q = R using auxiliary equations dx/P = dy/Q = dz/R.
Finding two independent integrals u=c1 and v=c2 is crucial for Lagrange's method.
The integral surface is derived by combining u, v with the given initial curve equation.
Homogeneous first-order ODEs are solved using the substitution y=vx (or x=vy).

Answer: Differential equations are fundamental in modeling various physical phenomena. This response addresses two distinct types of first-order differential equations: a partial differential equation (PDE) requiring Lagrange's method for its integral surface and an ordinary differential equation (ODE) solvable using the homogeneous equation technique or an integrating factor. Understanding these solution methodologies, as outlined in BMTC-132, is crucial for tackling such problems effectively. Lagrang...

Q4. No main question text, refer to sub-questions.

a)) Using Charpit's method, find the complete integral of the p.d.e.: 2xz - px² - 2qxy + pq = 0. (140 words)
b)) Using the method of undetermined coefficients, solve the differential equation: (D³ + 2D² – D – 2)y = e^x + x². (140 words)

Key Points:

Charpit's method finds complete integrals for first-order non-linear PDEs.
Charpit's auxiliary equations are used to derive relations for p and q.
The complete integral of 2xz - px² - 2qxy + pq = 0 is z = ay + b / (a - x²).
Homogeneous linear ODEs are solved by finding roots of the auxiliary equation.

Answer:

Q5. No main question text, refer to sub-questions.

a)) A particle falls from rest in a medium in which the resistance is λν² per unit mass, v being the velocity of the particle at time t. Prove that the distance fallen in time t is 1/λ Incosh (t√λg), where g is the acceleration due to gravity. (175 words)
b)) Solve: (y² + yz)dx + (z² + zx)dy + (y² – xy)dz = 0. (175 words)

Key Points:

Newton's second law for falling particle: m(dv/dt) = mg - mλv².
Integration of 1/(a²-x²) yields (1/(2a))ln|(a+x)/(a-x)|, crucial for velocity.
Velocity v(t) = dx/dt is integrated to find distance x(t) using ∫tanh(u)du = ln|cosh(u)|.
Homogeneous 3D differential equations Pdx+Qdy+Rdz=0 are solved by substitutions x=Xz, y=Yz.

Answer: The solution to the given differential equations involves applying fundamental concepts from mechanics and differential equations, as taught in BMTC-132. For the particle falling with resistance, Newton's second law leads to a separable first-order ordinary differential equation, which is solved by integration and application of initial conditions. The second problem is a first-order non-exact differential equation in three variables, which can be solved by recognizing its homogeneous nature and...

Q6. No main question text, refer to sub-questions.

a)) Solve: (D² + 5DD' + 5D'²)z = x sin (3x – 2y). (175 words)
b)) Solve: dx/(y² + yz + z²) = dy/(z² + zx + x²) = dz/(x² + xy + y²) (175 words)

Key Points:

Partial Differential Equations (PDEs) require finding Complementary Function (CF) and Particular Integral (PI).
CF for D² + 5DD' + 5D'² = 0 is f₁(y + m₁x) + f₂(y + m₂x) where m = (-5 ± √5)/2.
PI for x sin(ax+by) often uses complex exponential method (Im[ (1/F(D,D')) x e^(i(ax+by)) ]).
Systems of symmetric ODEs dx/P=dy/Q=dz/R are solved by finding two independent integrals.

Answer: Differential equations are fundamental in mathematics and engineering, used to model various phenomena. Solving them involves finding functions that satisfy the given equations. For partial differential equations (PDEs), the solutions often include arbitrary functions, while for ordinary differential equations (ODEs), they involve arbitrary constants. The provided questions require applying specific techniques for linear PDEs with constant coefficients and systems of symmetric first-order ODEs. ...

Q7. No main question text, refer to sub-questions.

a)) Using the method of variation of parameters, solve the equation d²y/dx² + a²y = sec ax. (140 words)
b)) Solve (p+q) (px + qy) = 1, using Charpit's method. (140 words)
c)) Solve: dy/dx = 1/(x+y+1) (70 words)

Key Points:

Variation of Parameters finds PI using complementary functions and Wronskian.
Wronskian (W) for y₁, y₂ is y₁y₂' - y₁'y₂, non-zero for linear independence.
Charpit's method solves first-order non-linear PDEs by finding p-q relation.
dz = pdx + qdy is integrated in Charpit's method after finding p and q.

Answer: The provided question requires solving three distinct differential equations using specific methods taught in the BMTC-132 course: Variation of Parameters for a second-order linear ODE, Charpit's method for a first-order non-linear PDE, and a substitution method for a first-order ODE. Each solution demonstrates a fundamental technique for handling different classes of differential equations, emphasizing the application of formulas and systematic steps outlined in the course material.

Q8. No main question text, refer to sub-questions.

a)) Solve: (D² – DD' – 2D) z = sin(3x + 4y) + e^(2x+y). (175 words)
b)) Use the method of variation of parameters to solve the following differential equation: y'' - 2y' + y = 12e^x/x³. (175 words)

Key Points:

CF for D(D-D'-2)z=0 is sum of solutions from factors Dz=0 and (D-D'-2)z=0.
PI for sin(ax+by) in PDE uses D²→-a², DD'→-ab, D'²→-b² rules, often requiring conjugate multiplication.
PI for e^(ax+by) in PDE uses D→a, D'→b substitution, provided the denominator is non-zero.
Variation of Parameters begins by finding homogeneous solutions y₁ and y₂ and their Wronskian W.

Answer:

Q9. No main question text, refer to sub-questions.

a)) Solve: x²y'' - 2xy' – 4y = x² + 2 ln x. (175 words)
b)) Solve the equation (7y-3x + 3)dy + (3y – 7x + 7)dx = 0. (175 words)

Key Points:

Cauchy-Euler equations: Solve homogeneous part using y=x^m for characteristic equation.
Particular solution (y_p) for Cauchy-Euler: Use Undetermined Coefficients for x^k and ln x terms.
Non-exact ODE reducible to homogeneous: Check exactness (∂M/∂y ≠ ∂N/∂x).
Coordinate Translation: Substitute x=X+h, y=Y+k by setting constant terms to zero.

Answer: This response comprehensively addresses the given differential equations, applying standard techniques taught in BMTC-132. Sub-question (a) involves a non-homogeneous Cauchy-Euler equation, which requires solving the homogeneous part and then finding a particular solution using the method of undetermined coefficients. Sub-question (b) presents a first-order non-exact differential equation reducible to a homogeneous form through a suitable translation of coordinates. Both solutions demonstrate st...

Q10. No main question text, refer to sub-questions.

a)) Using the method of undetermined coefficients, solve the equation d²y/dx² - 3 dy/dx + 2y = 4x². (175 words)
b)) Using Charpit's method, solve the equation zp² - y²p + y²q = 0. (175 words)

Key Points:

Method of Undetermined Coefficients solves non-homogeneous ODEs with specific RHS functions.
Complementary function `yc` is found by solving the auxiliary equation for homogeneous ODEs.
Particular integral `yp` form depends on the non-homogeneous term `R(x)` (e.g., polynomial, exponential, trigonometric).
Coefficients of `yp` are determined by substituting `yp` and its derivatives into the ODE and equating terms.

Answer: This response comprehensively addresses the given sub-questions from BMTC-132, focusing on the method of undetermined coefficients for ordinary differential equations and Charpit's method for partial differential equations. Each solution adheres to the specified word limits and demonstrates a step-by-step approach using relevant formulas and principles from the course material. For the ordinary differential equation, the solution involves first finding the complementary function by solving the ...

BMTC-132 Solved Assignment — January 2026 / July 2026

DIFFERENTIAL EQUATIONS | IGNOU Bachelor of Science (General) (CBCS) (BSCG)

BMTC-132—January 2026 / July 2026

DIFFERENTIAL EQUATIONS

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BMTC-132 Assignment Questions — January 2026

Q1. State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:

Q2. No main question text, refer to sub-questions.

Q3. No main question text, refer to sub-questions.

Q4. No main question text, refer to sub-questions.

Q5. No main question text, refer to sub-questions.

Q6. No main question text, refer to sub-questions.

Q7. No main question text, refer to sub-questions.

Q8. No main question text, refer to sub-questions.

Q9. No main question text, refer to sub-questions.

Q10. No main question text, refer to sub-questions.

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