Differential Equations — Class 12 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Differential Equations" — 8 important questions with detailed answers for CBSE board exam preparation.
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TL;DR: Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Differential Equations" — 8 important questions with detailed answers for CBSE boa…
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Key Questions Covered:
- Solve the differential equation dy/dx = 2x with the initial condition y(0) = 1.
- Solve the differential equation dy/dx + 2y = e^(-2x).
- Solve the differential equation (x + y) dx = (x + y + 1) dy.
- Solve dy/dx = (x + y)/(x - y).
- Solve the differential equation x(dy/dx) = y + x² cos x.
- Solve the differential equation d²y/dx² - 5(dy/dx) + 6y = 0.
- + 2 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| Solve the differential equation dy/dx = 2x with the initi… | ✓ Solved |
| Solve the differential equation dy/dx + 2y = e^(-2x). | ✓ Solved |
| Solve the differential equation (x + y) dx = (x + y + 1) dy. | ✓ Solved |
| Solve dy/dx = (x + y)/(x - y). | ✓ Solved |
| Solve the differential equation x(dy/dx) = y + x² cos x. | ✓ Solved |
| Solve the differential equation d²y/dx² - 5(dy/dx) + 6y = 0. | ✓ Solved |
Showing 6 of 8 questions
Q1: Solve the differential equation dy/dx = 2x with the initial condition y(0) = 1.
To solve dy/dx = 2x with y(0) = 1:
Step 1: Separate variables (already separated).
dy = 2x dx
Step 2: Integrate both sides.
∫ dy = ∫ 2x dx
y = x² + C
Step 3: Apply initial condition y(0) = 1.
1 = 0² + C
C = 1
Step 4: Write the particular solution.
y = x² + 1
Final Answer: y = x² + 1
Q2: Solve the differential equation dy/dx + 2y = e^(-2x).
To solve linear DE: dy/dx + 2y = e^(-2x):
Step 1: Identify the form.
This is dy/dx + P(x)y = Q(x)
where P(x) = 2 and Q(x) = e^(-2x)
Step 2: Find integrating factor.
I.F. = e^(∫ P(x) dx)
= e^(∫ 2 dx)
= e^(2x)
Step 3: Multiply the equation by I.F.
e^(2x) × dy/dx + 2e^(2x) × y = e^(2x) × e^(-2x)
e^(2x) × dy/dx + 2e^(2x) × y = 1
Step 4: Recognize left side as derivative of product.
d/dx[e^(2x) × y] = 1
Step 5: Integrate both sides.
e^(2x) × y = ∫ 1 dx
e^(2x) × y = x + C
Step 6: Solve for y.
y ...
Q3: Solve the differential equation (x + y) dx = (x + y + 1) dy.
To solve (x + y) dx = (x + y + 1) dy:
Step 1: Rearrange the equation.
(x + y) dx - (x + y + 1) dy = 0
Step 2: Use substitution.
Let u = x + y, then du = dx + dy
So dx = du - dy
Step 3: Substitute in the equation.
u(du - dy) - (u + 1) dy = 0
u du - u dy - u dy - dy = 0
u du - 2u dy - dy = 0
u du = (2u + 1) dy
Step 4: Separate variables.
u/(2u + 1) du = dy
Step 5: Integrate left side.
∫ u/(2u + 1) du = ∫ dy
For left integral, let 2u + 1 = t, then u = (t - 1)/2 and du = dt/2
= ∫ [(t - 1)/2]/t...
Q4: Solve dy/dx = (x + y)/(x - y).
To solve dy/dx = (x + y)/(x - y):
Step 1: Use substitution.
Let y = vx, then dy/dx = v + x(dv/dx)
Step 2: Substitute in the equation.
v + x(dv/dx) = (x + vx)/(x - vx)
v + x(dv/dx) = x(1 + v)/[x(1 - v)]
v + x(dv/dx) = (1 + v)/(1 - v)
Step 3: Rearrange.
x(dv/dx) = (1 + v)/(1 - v) - v
x(dv/dx) = [(1 + v) - v(1 - v)]/(1 - v)
x(dv/dx) = [1 + v - v + v²]/(1 - v)
x(dv/dx) = (1 + v²)/(1 - v)
Step 4: Separate variables.
(1 - v)/(1 + v²) dv = dx/x
Step 5: Integrate both sides.
∫ (1 - v)/(1 + v²) dv =...
Q5: Solve the differential equation x(dy/dx) = y + x² cos x.
To solve x(dy/dx) = y + x² cos x:
Step 1: Rewrite in standard form.
dy/dx - y/x = x cos x
dy/dx + (-1/x)y = x cos x
Step 2: Find integrating factor.
I.F. = e^(∫ -1/x dx) = e^(-ln|x|) = 1/x
Step 3: Multiply equation by I.F.
(1/x)(dy/dx) - y/x² = cos x
Step 4: Recognize left side as derivative.
d/dx[y/x] = cos x
Step 5: Integrate both sides.
y/x = ∫ cos x dx
y/x = sin x + C
Step 6: Solve for y.
y = x sin x + Cx
Final Answer: y = x(sin x + C)
Q6: Solve the differential equation d²y/dx² - 5(dy/dx) + 6y = 0.
To solve second-order linear ODE: d²y/dx² - 5(dy/dx) + 6y = 0:
Step 1: Write the characteristic equation.
Assuming y = e^(mx), we get:
m² - 5m + 6 = 0
Step 2: Solve the characteristic equation.
Using factorization:
(m - 2)(m - 3) = 0
m = 2 or m = 3
Step 3: Write the general solution.
Since we have two distinct real roots, the general solution is:
y = C₁e^(2x) + C₂e^(3x)
where C₁ and C₂ are arbitrary constants.
Final Answer: y = C₁e^(2x) + C₂e^(3x)
Showing 6 of 8 questions. Visit the full page for complete solutions.
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